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ALGEBRA  FOR  SCHOOLS 


GEORGE   W.    EVANS 

Instructor  in  Mathematics  in  the  English  High  School 
Boston,  Mass. 


NEW   YORK 

HENRY   HOLT   AND   COMPANY 
1899 


Copyright,  1899, 

BV 

HENRY  HOLT  &  CO. 


ROBERT   DRUMMOND,    PRINTER.    NEW  YORK 


A/^-3 


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PREFACE. 

In  the  arrangement  of  topics  in  this  book  an  effort  has 
been  made  to  preserve  the  pupil  from  the  besetting  sin  of 
conceiving  algebraic  operations  as  a  species  of  legerdemain. 
This  end  could  best  be  secured,  it  seemed  to  me,  by  making 
practical  problems  the  point  of  departure,  initially  and  at 
each  new  turn  of  the  subject.  With  a  concrete  case  in 
mind,  the  pupil  can  hardly  fail  to  perceive  not  only  the 
need  for  the  process  that  he  is  set  to  study,  but  also  its 
rational  basis  and  its  economy.  In  this  larger  appeal  to 
the  practical  sense  it  will  be  found  that  there  is  no  slighting 
of  mental  dexterity,  no  injurious  deviation  from  accepted 
methods,  and  certainly  no  sacrifice  of  mathematical  rigor. 

If  the  arrangement  of  chapters  here  adopted  is  not 
acceptable,  it  is  entirely  feasible  to  take  the  topics  in  the 
traditional  order.  The  index  will  facilitate  this  rearrange- 
ment. 

Whether  or  not  teachers  agree  with  me  in  respect  to  the 
ordering  of  topics  or  the  method  of  attack  which  I  have 
found  fruitful  in  class,  they  are  sure  to  appreciate  the  very 
large  collection  of  examples — some  thirty-five  hundred  all 
told — which  are  not  reprinted  from  other  text-books. 

In  the  following  features  also  I  think  the  book  possesses 
advantages : 

1.  The  careful  classification  of  problems,  so  that  due  em- 
phasis may  be  given  to  the  several  types  of  equations  arising 
from  them, 

iii 


ivi30B0»7 


iv  PREFACE, 

2.  The  insistence  upon  a  scheduled  explanation  of  steps 
in  the  reduction  of  equations;  and  the  clearness  and  brev- 
ity of  reference  and  explanation  obtained  by  denoting  an 
equation  by  its  serial  number  enclosed  in  a  circle. 

3.  The  introduction  of  supplementary  sets  of  constants 
for  some  important  problems,  as  on  pages  28  and  39  and 
in  Chapter  V. 

4.  The  thorough  study  of  literal  equations  and  general- 
ized problems;  and  especially  of  literal  quadratics. 

5.  The  prominence  given  to  the  solution  of  equations  by 
factoring  as  a  fundamental  method;  and  the  treatment  of 
''  completing  the  square^''  as  a  method  of  factoring. 

6.  The  treatment  of  Eules  and  Formulae  in  Chapter  II, 
and  of  Theorems  and  Identities  in  Chapter  IV. 

7.  The  separation  of  Elimination  into  two  parts  suitable 
for  linear  systems  and  for  linear-quadratic  pairs  respectively ; 
the  discussion  of  simultaneous,  independent,  and  consistent 
equations;  the  ^^  Equations  of  the  New  Set,'^  page  160. 

8.  The  treatment  of  H.  C.  E.  by  elimination  of  highest 
and  lowest  terms. 

Much  of  the  book  has  been  used,  in  manuscript  and  in 
proof,  for  class- work,  but  I  can  hardly  hope  that  all  mis- 
takes have  been  corrected :  notice  of  remaining  errors  will 
be  gratefully  received. 

Eor  friendly  aid  and  suggestion  I  am  indebted  to  several 
of  my  colleagues;  and  especially  to  Mr.  P.  E.  Gartland, 
who  scrutinized  for  me  every  letter  of  the  proofs. 

G.  W.  E. 

English  High  School,  Boston, 
February  3,  1899. 


CONTENTS. 


PAGE 

I.  The  first  use  of  algebra:    abbreviation  of  expla- 
nations        1 

II.  The  second  use  op  algebra:  abbreviation  of  rules    41 

III.  Transformations 56 

IV.  The  third  use  of  algebra:  theorems  and  identi- 

ties . .   88 

V.  Factorable  equations 104 

VI.  The  first  method  of  elimination:  equations  of  the 

FIRST  DEGREE 131 

VII.  The  second  method  of  elimination:  linear-quad- 
ratic PAIRS. 163 

VIII.  Multiplication  of  fractions;  highest  common  fac- 
tor   190 

IX.  Lowest  common  multiple;  summation  of  fractions .  212 

X.  Indices;  surds;  roots 239 

XI.  Literal  equations;  generalization 274 

XII.  Proportion  and  variation 299 

XIII.  The  progressions 310 

XIV.  Detached  coefficients;   the  binomial  theorem.  . . .  328 
XV.  Logarithms 348 

Table  of  Logarithms 363 

XVI.  Supplementary  PROBLEMS  FOR  practice  AND  review.  366 


ALGEBRA. 

CHAPTER  I. 
THE  FIRST  USE  OF  ALGEBRA. 

1.  Algebra  is  a  method  of  abbreviating  the  explana- 
tions of  problems  in  arithmetic.  It  is  also  used  to  abbre- 
viate the  statement  of  rules  and  the  demonstration  of 
theorems. 

2.  This  chapter  illustrates  the  first  use  of  Algebra:  for 
abbreviating  the  explanation  of  problems  in  arithmetic. 
Some  problems  are  here  given  as  examples;  the  answers  to 
many  of  them  are  evident  at  a  glance.  The  pupil  must  re- 
member that  he  is  not  expected  to  shorten  the  arithmetical 
work,  for  he  cannot  do  that.  He  is  to  put  down  the  ex- 
planation briefly  and  systematically.  When  that  is  learned 
he  will  find  that  he  can  do  complicated  problems  with 
greater  ease,  because  he  can  put  down  in  black  and  white 
as  he  goes  along  the  successive  steps  of  his  reasoning. 

3.  Model  A. — A  father  is  6  times  as  old  as  his  son,  and 
their  united  ages  are  42  years;  find  the  age  of  each. 

Explanation, 

(1)  The  father's  age  -j-  the  son's  age  =  42  years. 

(2)  The  father's  age  =  6  X  the  son's  age. 

(3)  6  X  the  son's  age  +  ^^6  son's  age  =  42  years. 

(4)  7  X  the  son's  age  =  42  years. 

(5)  The  son's  age  =  6  years. 

(6)  The  father's  age  =  36  years. 


2  ALGEBRA, 

Abbreviated  Explanation, 

Let  5  stand  for  the  number  of  years  in  the  son's 
age ;  then  6X5  will  stand  for  the  number  of  years 
in  the  father^s  age. 

0    6X5  +  5  =  42 
©  7  X  5  =  42 

®  5  =  6 

®  6  X  5  =  36 

EXERCISE    I. 

1.  John  is  twice  as  old  as  Henry,  and  the  sum  of  their » 
ages  is  21  years.     Find  the  age  of  each. 

2.  John  and  Will  wish  to  divide  36  cents  so  that  John 
shall  have  twice  as  much  as  Will.  How  much  will  each 
have? 

3.  Paid  $220  for  a  horse  and  saddle,  and  the  horse  cost 
10  times  as  much  as  the  saddle.     Cost  of  each  ? 

4.  A  sidewalk  laid  with  a  flagstone  and  a  curbstone  is  6 
feet  wide;  the  flagstone  is  11  times  as  wide  as  the  curb- 
stone.    Width  of  the  flag  and  of  the  curb  ? 

5.  The  electoral  vote  for  the  state  of  Pennsylvania  is  5 
times  that  for  the  state  of  West  Virginia.  Both  count  36 
electoral  votes.     Each  counts  how  many  ? 

6.  There  are  7  times  as  many  sheep  as  lambs  in  a  pasture, 
and  in  all  there  are  96.     Number  of  each  ? 

7.  Pole  9  feet  long  partly  peeled  of  bark;  bark  part 
twice  as  long  as  bare  part.     Find  length  of  each  part. 

8.  It  takes  a  man  4  times  as  long  to  run  the  first  three- 
quarters  of  a  measured  mile  as  to  run  the  last  quarter;  he 
goes  the  whole  mile  in  6  minutes.  How  long  for  the  last 
quarter  ? 

9.  Mercury  weighs  13  times  as  much  as  water;  a  quart 
of  mercury  and  a  quart  of  water  weigh  in  all  28  pounds. 
Weight  of  each  ? 


THE  FIRST  USE  OF  ALGEBRA.  3 

10.  A  bag  of  oats,  hung  on  one  end  of  a  steelyard  beam, 
weighs  20  times  as  much  as  the  iron  weight  attached  to  the 
other  end  of  the  beam ;  the  total  downward  pull  on  the 
steelyard  is  52J  pounds.  Find  the  weight  of  the  bag  of 
oats. 


4.  An  equation  is  a  statement  that  two  things  are  equal. 

5.  The  algebraic  abbreviation  for  any  quantity,  or  for 
any  combination  of  quantities,  is  called  an  algebraic  ex- 
pression. 

6.  It  takes  two  algebraic  expressions  to  form  an  equa- 
tion. The  two  expressions  said  to  be  equal  are  called,  re- 
spectively, the  first,  or  left-hand  member,  and  the  second, 
or  right-hand  member,  of  the  equation. 

7.  The  parts  of  an  algebraic  expression  separated  by  the 
signs  +  aiid  "~  ^^'^  called  the  terms  of  the  expression. 

8.  In  order  to  fix  the  student's  mind  on  the  nature  of  the 
processes  by  which  his  equations  are  obtained,  it  is  well  for 
him  to  indicate  beside  each  equation  how  it  arose  from  the 
preceding  equations.  The  additional  trouble  is  slight  and 
the  scheduled  '*  explanation  "  is  logically  complete. 

9.  Model  B. — In  a  certain  collection  of  insects  there  are 
twice  as  many  butterflies  as  moths;  of  both  there  are  39. 
How  many  of  each  ? 

Let  X  =  the  number  of  moths. 

Then  2x*  =  the  number  of  butterflies. 

0  2.T  +    X  =  39 

(D  3x  =  39  same  as  0 

@  x  =  U  (D  -^  3 

®  2x  =  2Q  (D  X  2 

Ans.  13  moths;  26  butterflies. 

^  When  two  quantities  are  written  together  without  any  sign  be- 
tween them,  multiplication  is  indicated.  For  representing  in  any  prob- 


4  ALGEBRA. 

EXERCISE  II, 

1.  One  man  is  twice  as  heavy  as  another,  and  both  weigh 
339  pounds.     Weight  of  each  ? 

2.  The  number  of  cows  and  sheep  in  a  certain  farmyard 
is  75,  and  there  are  4  times  as  many  sheep  as  cows.  Num- 
ber of  each  ? 

3.  The  doctor  has  twice  as  many  books  as  the  minister, 
and  they  both  have  2100.     How  many  books  has  each  ? 

4.  In  the  dog-show  are  36  St.  Bernards,  twice  as  many 
long-haired  as  short-haired.     Number  of  each  kind  ? 

5.  A  locomotive  weighs  6  times  as  much  as  a  car,  and 
both  weigh  14  tons.     Weight  of  each  ? 

6.  A  man  is  twice  as  heavy  as  a  boy,  and  both  weigh  200 
pounds.     Weight  of  each  ? 

7.  A's  house  cost  2^  times  as  much  as  B^s  house,  and 
both  cost  $7000.     Cost  of  each  ? 

8.  Paid  $100  for  two  lots  of  tiles;  one  lot  cost  4  times  as 
much  as  the  other.     Cost  of  each  lot  ? 

9.  Walked  4  times  as  far  in  the  afternoon  as  I  did  in  the 
forenoon ;  all  day  together  I  walked  15  miles.  How  far  in 
the  forenoon  ? 

10.  It  costs  4  times  as  much  to  go  from  New  York  to 
Chicago  as  from  Boston  to  New  York;  from  Boston  to 
Chicago  it  costs  $25.  How  much  from  Boston  to  New 
York? 

11.  The  president  of  a  stock  company  owns  twice  as 
many  shares  in  it  as  his  brother;  and  both  own  165.  How 
many  does  each  own  ? 

12.  A  man  had  a  big  bill  in  his  pocket,  and  asked  the 
price  of  a  house  and  barn;  he  found  that  the  house  cost 

lem  numbers  of  which  the  values  are  not  given,  the  latter  letters  of 
the  alphabet  are  generally  used,  most  often  x.  These  are  matters  of 
universal  agreement  among  writers  and  students  of  Algebra  (conven- 
tions). 


THE  FIRST  USJE  OF  ALOEBRA.  5 

7  times  the  value  of  the  bill,  and  the  barn  3  times  the  value 
of  the  bill,  both  coming  to  $5000.  Find  cost  of  house  and 
of  barn. 

13.  Three  times  as  many  barrels  in  one  cellar  as  in  an- 
other; in  the  store  above  half  as  many  as  in  both  cellars. 
In  the  whole  building  216  barrels.  How  many  in  each 
part  ? 

14.  A  fortress  is  garrisoned  by  5200  men;  and  there  are 
9  times  as  many  infantry  and  3  times  as  many  artillery  as 
cavalry.     How  many  are  there  of  each  ? 

Valuation  Problems. 

10.  Model  C— Settled  an  account  of  $48  with  $2  and  $5 
bills,  using  twice  as  many  5's  as  2's.  Find  the  number  of 
bills  of  each  denomination. 

Let  X  =  the  number  of  $2  bills. 

Then  2x  =  the  number  of  $5  bills. 

®  lOo;  +    2x  =  4:S^ 

(2)  12iz;  =  48  same  as  0 

(D  a:  =  4  ©  -^  12 

®  2:?:  =  8  (D  X  2 

A71S.  Four  $2  bills;  eight  $5  bills. 

EXERCISE  in. 

How  many  bills  of  each  kind  must  I  use  to  settle  the 
following  accounts  in  the  manner  described  for  each  ac- 
count ? 

1.  To  pay  $88  I  use  5's  and  2's  only,  3  times  as  many  2's 
as  5's. 

2.  To  pay  $72  I  use  5's  and  2's  only,  2  times  as  many  5's 
as  2's. 

*  The  statement  is  about  tlie  number  of  dollars  in  the  values  of 
the  bills,  not  about  the  number  of  bills;  the  value  of  the  $5  bills  is 
5  times  their  number,  5  X  2aj  or  lOiC. 


6  ALGEBRA. 

3.  To  pay  $42  I  use  lO's  and  2^s  only,  twice  as  many 
2's  as  lO's. 

4.  To  pay  $80  I  use  lO's  and  5's  only,  twice  as  many 
5's  as  lO's. 

5.  To  pay  $26  I  use  lO's  and  l^s  only,  3  times  as  many 
I's  as  lO's. 

6.  To  pay  $45  I  use  2's  and  l^s  only,  7  times  as  many  2's 
as  I's. 

7.  To  pay  $78  I  use  5's  and  I's  only,  5  times  as  many 
5's  as  I's. 

8.  To  pay  $84  I  use  lO's  and  2's  only,  4  times  as  many 
lO's  as  2's. 

9.  To  pay  $75  I  use  lO's  and  5's  only,  7  times  as  many 
lO's  as  5's. 

10.  To  pay  $183  I  use  lO's  and  I's  only,  6  times  as  many 
lO's  as  I's. 

11.  Five  horses  and  four  donkeys  weigh  all  together  9600 
pounds.  Each  horse  weighs  4  times  as  much  as  a  donkey. 
Find  the  weight  of  each  animal. 

12.  Bought  apples  for  39  cents;  twice  as  many  red  ones 
for  5  cents  apiece  as  green  ones  for  3  cents  apiece.  Num- 
ber of  each  ? 

13.  Engineers  examined  a  bridge  that  broke  down  under 
the  weight  of  a  crowd  of  people,  and  decided  that  the 
breaking  strain  was  36  tons.  Supposing  that  there  were 
4  times  as  many  men  as  women,  the  average  weight  being 
150  pounds  for  a  man  and  120  pounds  for  a  woman,  what 
was  the  number  of  each  ? 

14.  Tea  of  first  quality  at  60  cents  per  pound,  and  3 
times  as  much  tea  of  second  quality  at  45  cents  per  pound, 
came  to  $13.65.     Number  of  pounds  of  each? 

15.  Mules  at  $40  apiece,  and  7  times  as  many  horses  at 
$125  apiece,  cost  $2745.     Number  of  animals  in  the  drove  ? 

16.  Drove  westward  in  a  buggy  for  10  hours,  then  rode 


THE  FIRST   USE  OF  ALGEBRA,  7 

in  a  train  5  times  as  fast  for  two  days  and  nights ;  the  total 
journey  was  1500  miles.     How  fast  did  the  buggy  go  ? 

17.  Lead  weighs  four  times  as  much  as  marble;  three 
leaden  globes  and  two  marble  globes,  all  of  the  same  size, 
weigh  56  pounds.     AVeight  of  each  globe  ? 

18.  A  team  is  made  up  of  oxen  and  mules,  one  of  each 
in  a  pair;  supposing  that  an  ox  can  pull  4  times  as  much 
as  a  mule,  and  the  whole  team  is  used  to  pull  30  tons,  how 
much  of  this  load  do  the  oxen  pull  ? 

19.  A  team  consisting  of  2  oxen  and  5  mules,  in  which 
each  ox  pulls  twice  as  much  as  a  mule,  is  used  to  raise  a 
safe  weighing  18  tons.  What  would  be  the  strain  on  each 
trace  of  each  mule  ?  What  would  be  the  strain  on  the  ox- 
chain  ? 

20.  A  horizontal  iron  beam  weighing  one  ton  is  sus- 
pended by  3  chains,  2  at  one  end  and  1  at  the  other. 
What  is  the  pull  on  each  chain  ? 

21.  Paid  91  cents  with  dimes  and  three-cent  pieces,  of 
each  the  same  number.     Number  of  each  ? 

22.  Three  times  as  many  nickels  as  two-cent  pieces  in 
my  purse;  in  all  51  cents.     Number  of  each  ? 

23.  Paid  $1.80  with  quarters,  dimes,  and  nickels;  twice 
as  many  dimes  and  3  times  as  many  nickels  as  quarters. 
Number  of  each  ? 

24.  Bought  twice  as  much  coffee  at  33  cents  as  tea  at  54 
cents,  and  paid  in  all  $2.40.  How  many  pounds  of  each 
were  bought  ? 

25.  Bought  horses  at  $120,  sheep  at  $14,  and  chickens 
at  50  cents;  3  times  as  many  sheep  as  horses,  7  times  as 
many  chickens  as  sheep.  All  cost  $517.50.  How  many 
of  each  ? 

26.  Two  travellers  started  towards  each  other  from  oppo- 
site ends  of  a  straight  road  132  miles  long;  one  went  4 
miles  an  hour,  the  other  7.     How  long  before  they  met  ? 


8  ALGEBRA, 

27.  The  crew  of  a  towboat  consists  of  an  engineer,  2 
firemen,  3  deck-hands,  and  a  cook,  whose  wages  are  re- 
spectively $2,  $1.50,  $1,  and  $1.20  per  day.  For  a  certain 
voyage  the  pay-roll  was  $73.60.     How  many  days  out  ? 

28.  In  a  certain  mill  are  employed  men  at  an  average 
wage  of  $1.17  per  day,  5  times  as  many  women  at  an 
average  wage  of  63  cents  per  day,  and  twice  as  many  chil- 
dren (as  men)  at  an  average  wage  of  31  cents  per  day. 
Tlie  weekly  pay-roll  of  this  mill  is  $296.40.  How  many 
men,  women,  and  children  are  employed  in  it  ? 

29.  Three  families,  with  8,  5,  and  3  members  respect- 
ively, divide  among  themselves  a  square  mile  of  land,  so 
that  there  are  equal  shares  for  all  the  individuals.  What 
is  the  share  of  each  family  ? 

30.  A  sidewalk  is  laid  with  3  flags  and  a  curb,  the  flags 
being  each  7  times  as  wide  as  the  curb.  The  width  of  the 
sidewalk  is  11  feet.     Find  the  width  of  flag  and  of  curb. 

Difference  of  the  Unknown  Numbers  Given. 

11.  Model  D. — A  father  is  50  pounds  heavier  than  his 
son,  and  both  weigh  248  pounds.     Weight  of  each  ? 
Let  X  =  the  number  of  pounds  the  son  weighs. 
Then  x -\-  60  =  the  number  of  pounds  the  father  weighs. 
©  a:  +  a;  +  50  =  248 
@        2:r  +  50  =  248  same  as  ® 

(D  2:?:  =  198  ©  -  50 

®  a;  =  99  (D  -f-  2 

(D  a;  +  50  ==  149  ®  +  50 

Ans,  Father  149  pounds;  son  99  pounds. 

EXERCISE   IV. 

1.  Two  cannon-balls  and  a  six-pound  weight  balanced 
on  a  scale  50  pounds.  What  was  the  weight  of  each 
canno.n-ball  ? 


THE  FIRST  USE  OF  ALGEBRA.  9 

2.  A  marketman  sold  11  sheepskins,  and  then  lost  $2 
of  the  purchase-money;  he  had  $7.90  left.  How  much 
did  a  sheepskin  sell  for  ? 

3.  On  a  bill  of  1181  part  of  the  account  is  paid  with 
equal  numbers  of  $10,  $5,  and  $2  bills,  and  a  check  for  $28 
is  given  for  the  remainder.  How  many  bills  of  each  kind 
were  used  ? 

4.  Two  mules  and  a  horse  were  bought  for  $400,  and  a 
horse  cost  $40  more  than  a  mule.  Cost  of  each  ani- 
mal ? 

5.  Two  bucketfuls  of  water  fill  a  15 -gallon  tub  all  but  5 
quarts.     What  is  the  capacity  of  the  bucket  ? 

6.  A  man  starts  on  a  journey  of  163  miles;  he  walks  13 
hours,  rides  twice  as  fast  for  20  hours,  and  finally  has  to 
stop  4  miles  short  of  his  journey's  end.  How  fast  did  he 
walk  ? 

7.  A  grocer  mixes  tea  that  costs  him  20  cents  a  pound 
with  3  times  as  much  tea  that  costs  him  38  cents  a  pound, 
and  sells  the  mixture  for  $10,  clearing  a  profit  of  62  cents. 
Number  of  pounds  of  each  ? 

8.  My  farm  is  three  times  as  large  as  the  one  next  to  it, 
and  both  together,  with  a  ten-acre  lot  across  the  road,  just 
make  a  quarter-section — 160  acres.     Size  of  each  farm  ? 

9.  Two  trains  start  from  opposite  ends  of  the  same  13- 
mile  track,  one  going  20  miles  an  hour  and  the  other  24. 
They  stop  when  2  miles  apart.  How  long  were  they 
going  ? 

10.  Quarters  in  one  pile,  twice  as  many  dimes  in  another, 
and  $1.13  besides — in  all  $6.08.  How  many  quarters  and 
dimes  ? 

11.  Bought  peas  at  59  cents  a  peck,  and  1  peck  more  of 
beans  at  30  cents  a  peck;  paid  25  cents  for  lettuce — in  all 
$5.     How  many  pecks  of  peas  and  of  beans  ? 

12.  Bought  sugar  at  5  cents,  3  times  as  much  coffee  at 


10  ALGEBRA. 

50  cents,  and  paid  $1.83  for  a  ham.     The  whole  bill  was 
$12.68.     How  much  coffee  ? 

13.  Sold  a  house;  then  sold  5  more  at  double  that  price; 
then  sold  35  tons  of  coal  at  $5  per  ton.  Eeceived  all  together 
$64,448.     Price  of  first  house  sold  ? 

14.  Three  times  as  many  nickels  as  half-dollars;  in  all 
$4.55.     Number  of  each  ? 

15.  Three  times  as  many  dimes  as  nickels,  5  times  as 
many  three-cent  pieces  as  dimes;  in  all  $5.60.  Number 
of  each  ? 

16.  Three  times  as  many  dimes  as  dollars,  4  times  as 
many  cents  as  dimes,  5  times  as  many  nickels  as  cents;  and 
$17.68  all  together.     Number  of  nickels  ? 

17.  I  have  10  cents  more  than  my  uncle,  and  we  both 
have  $2.90.     How  much  has  each  ? 

18.  Suppose  each  of  the  36  boys  in  a  class  has  the  same 
sum,  and  the  teacher  has  15  cents  more  than  all  of  them 
together.  All  the  money  counts  up  $79.35.  How  much 
has  the  teacher  ? 

19.  Three  times  as  many  lO's  as  2^s,  and  $7.35  besides, 
make  $167.35.     Number  of  2'o  ? 

20.  Seven  times  as  many  I's  as  5's,  and  $1.45  besides, 
make  $37.45.     Number  of  I's  ? 

21.  Three  times  as  many  2's  as  5's,  4  times  as  many  lO's 
as  2's,  and  $11.11  besides,  make  $666.11.     Number  of  2's  ? 

22.  Eode  6  miles  an  hour,  then  walked  7  times  as  long 
at  4  miles  per  hour,  then  went  23  miles  by  train;  91  miles 
in  all.     How  long  did  I  walk  ? 

23.  Made  a  mixture  of  water,  3  times  as  much  wine  at 
$1.20  per  quart,  twice  as  much  bark  extract  as  wine  at 
75  cents  per  quart,  and  one-fourth  of  a  pint  of  beef  extract 
costing  $3.25.  Sold  it  all  at  a  profit  of  $5  and  received 
$24.45.     Number  of  quarts  of  bark  extract  ? 

24.  Five  times  as   many  nickels  as  two-cent  pieces,   5 


THE  FIRST   USE  OF  ALGEBRA  II' 

times  as  many  dimes  as  nickels,  7  times  as  many  quarters 
as  nickels,  twice  as  many  dollars  as  dimes,  and  a  check 
for  $53.92,  making  $300.     Number  of  dollars  ? 

Reduction  of  Equations. 

12.  Consider  the  problem:  A  is  twice  as  old  as  B;  22 
years  ago  he  was  3  times  as  old  as  B.  What  are  their  ages 
now  ? 

Let  X  =  B's  age  now.  Twenty-two  years  ago  A's  age 
was  2x  —  22  and  B's  age  was  x  —  22.  So  the  equation  is 
2a;  -  22  =  ^x  -  22). 

Expressions  like  3(^  —  22)  are  new  to  the  pupil. 

Suppose  a  farmer  contracts  to  deliver  3  bushels  of  oats 
and  5  bushels  of  barley  every  day.  To  get  the  number  of 
bushels  of  grain  delivered  in  several  days  he  multiplies  not 
only  the  quantity  of  oats  but  also  the  quantity  of  barley 
by  the  number  of  days. 

Suppose  a  farmer  receives  daily  x  bushels  of  grain,  and 
delivers  daily  3  bushels.  The  increase  of  grain  in  his  store 
for,  say,  7  days  would  be  found  thus: 

c?;  bushels  daily  for  7  days,    ....     received  72; ; 
3  bushels  daily  for  7  days,     ....     delivered  21. 

Subtracting  the  amount  delivered  from  the  amount  re- 
ceived, Ix  —  21  is  what  the  daily  increase  of  a;  —  3  amounts 
to  in  7  days. 

Consideration  of  similar  cases  will  make  the  following 
principle  evident. 

13.  Whenever  an  expression  of  two  or  more  terms  is 
multiplied,  each  term  of  that  expression  separately  must 
be  multiplied. 

14.  The  equation  2x — 22  =  3(2;  —  22)  now  becomes 
22;  —  22  =  3a;  —  66,  the  only  change  being  a  change  of 
form,  not  of  value. 

This  equation  22;  —  22  =  32;  —  66  differs  from  those  we 


12  ALGEBRA. 

have  been  used  to  in  that  it  has  negative  terms  on  each  side 
of  the  equation,  that  is,  terms  to  be  subtracted. 

The  left  side  is,  not  equal  to  ^x,  but  just  22  short  of 
it.     In  the  same  way  the  right  side  is  66  short  of  ^x. 

If  22  is  added  to  each  side  of  the  equation,  the  shortage 
on  the  left  side  will  disappear,  but  the  right  side  will  still 
be  44  short  of  3x.  That  is,  the  equation  will  be  2x  =  3^  — 
44.  This  shortage  also  disappears  when  44  is  added  to  each 
side,  giving  2^  +  ^^  =  3a;,  an  equation  like  many  we  have 
been  solving. 

But  if  66  is  added  at  first  the  entire  shortage  on  the 
right  is  made  up;  and  on  the  left  it  only  takes  22  out  of 
the  66  to  make  up  that  shortage,  leaving  44  to  be  added  to 
the  complete  value  of  "Hx,  The  shortest  way,  then,  when 
there  are  two  similar  shortages,  is  to  add  the  larger. 


Shortages  and  Multiplications. 

15.  Model  E. — A  is  twice  as  old  as  B;  22  years  ago  he 
was  three  times  as  old  as  B.     What  are  their  ages   now  ? 
Let  X  =  B's  age  now ;  then  A's  age  =  2a;. 
©  2a;  -  22  =  3(a;  -  22) 
©  2a;  —  22  =  3a;  —  66,  same  as  ® 
(D  2a;  +  44  =  3a;  ©  +  66 

®  44  =  a;  ©  —  2a; 

®  88  =.  2a;  ©  X  2 

Ans.  A^s  age  88;  B's  age  44. 

EXERCISE  V. 

1.  A  is  5  times  as  old  as  B,  and  5  years  hence  will  be 
only  3  times  as  old  as  B.     Ages  now  ? 

2.  Eleven  years  ago  A  was  4  times  as  old  as  B  and  in  13 
years  from  now  he  will  be  only  twice  as  old.     Ages  now  ? 

3.  Rice  costs  2  cents  a  pound  more  than  sugar;  3  pounds 


THE  FIRST  USE  OF  ALGEBRA,  13 

of  sugar  and  10  pounds  of  rice  come  to  $2.40.     Cost  of 
each  ? 

4.  I  can  walk  3  miles  more  in  a  forenoon  than  in  an 
afternoon;  and  between  Monday  noon  and  Friday  night  I 
can  walk  75  miles.     How  far  can  I  walk  in  one  afternoon  ? 

5.  A  wall  13|:  feet  high  is  built  with  seven  courses  of 
foundation-stone  and  41  courses  of  brick  ;  the  courses  of 
stone  are  9  inches  thicker  than  the  brick.  What  is  the 
thickness  of  the  foundation-stones  ? 

6.  A  wall  is  laid  with  bricks  of  two  thicknesses,  3  inches 
and  5  inches ;  there  are  three  more  courses  of  the  thicker  kind 
than  of  the  thinner,  and  the  wall  is  9  feet  3  inches  high. 
Number  of  courses  of  each  ? 

7.  Paid  12.20  in  quarters  and  nickels  ;  two  more  nickels 
than  quarters.     Number  of  each  kind  of  coin  ? 

8.  A  boy  starts  from  his  room  in  college  at  noon  one  day 
to  walk  to  his  home,  41  miles  off  ;  at  two  o^clock  of  the 
same  day  the  father  starts  on  horseback  for  the  college, 
riding  3  miles  per  hour  faster  than  the  son  can  walk  ;  they 
meet  at  5  p.m.     How  fast  can  the  boy  walk  ? 

9.  A  pile  of  27  cannon-balls,  in  three  sizes,  weighs  254 
pounds  ;  10  of  them  are  twice  as  heavy  as  the  13  lightest  ; 
and  the  others  are  each  by  2  pounds  the  heaviest  in  the 
pile.     Find  the  weight  of  each  size. 

10.  A  tank  holding  5940  gallons  is  filled  in  3  hours 
by  three  pipes,  the  first  of  which  carries  twice  as  much, 
and  the  other  3  gallons  less,  per  minute,  than  the  third. 
Number  of  gallons  per  minute  through  each  pipe  ? 

11.  A  is  three  times  as  old  as  B  ;  15  years  ago  he  was 
five  times  as  old  as  B.     What  are  their  ages  now  ? 

Find  the  number  of  coins  of  each  denomination  used  in 
the  folloiuing  payments : 

12.  Paid  39  cents  with  three-cent  pieces  and  five-cent 
pieces;  5  more  3's  than  5*s, 


14  ALGEBRA, 

13.  Paid  85  cents  with  nickels  and  dimes;  2  more  nickels 
than  dimes. 

14.  Paid  $2.05  with  quarters  and  nickels;  1  fewer  nickels 
than  quarters. 

15.  Paid  $4.75  with  quarters  and  halves;  5  fewer  quarters 
than  halves. 

16.  Paid  $1.45  with  quarters  and  dimes;  4  more  dimes 
than  quarters. 

17.  Paid  68  cents  with  dimes  and  two-cent  pieces;  10 
more  2^s  than  dimes. 

18.  Paid  $2.30  with  quarters  and  three-cent  pieces;  2 
more  3's  than  quarters. 

19.  Paid  $1.09  with  two-cent  pieces  and  three-cent 
pieces;  3  more  3's  than  2''s. 

20.  Paid  $1.01  with  dimes  and  three-cent  pieces;  12 
more  3's  than  dimes. 

21.  Paid  $6.40  with  nickels  and  halves;  4  more  halves 
than  nickels. 

REVIEW. 

I.  What  is  the  use  of  Algebra  ?  What  is  an  equation  ? 
What  is  a  term  ? 

II.  State  the  first  and  second  members,  and  the  first, 
second,  and  third  terms,  in  the  following  equations: 

i.  3:r  -  5  +  lOiT  =  7  -  X. 

2.  X  —  y  +  5  —  3y  +  a, 

S.  x-x  +  l  +  bx  —  y  +  6=x+5x+l. 

4.  4x=z3  +  5x-  y  +  Q. 

III.  Solve  the  following  problems: 

1,  Thirty  horses  and  40  mules  weigh  54  tons;  on  an 
average,  each  horse  weighs  100  pounds  more  than  a  mule. 
Average  weight  of  the  two  kinds  of  animals  ? 

2,  There  are  in  a  purse  three  times  as  many  nickels  as 
dimes,  and  in  all  $1.50.     How  many  nickels  ? 


THE  FIRST  USE  OF  ALGEBRA.  15 

S,  Four  cows,  3  calves,  and  10  sheep  cost  $168;  a  cow 
costs  five  times  as  much  as  a  (;alf,  and  a  calf  costs  twice  as 
much  as  a  sheep.     Cost  of  each  ? 

^.*  A  man  wishes  to  pay  $35  with  dollars,  halves,  and 
quarters,  of  each  an  equal  number.     Number  of  each  ? 

5.  A  grocer  mixes  tea  that  cost  him  25  cents  a  pound 
with  four  times  as  much  tea  that  cost  him  30  cents  a  pound, 
and  sells  the  mixture  for  $16,  clearing  a  profit  of  $1.50. 
Number  of  pounds  of  each  ? 

Rule  for  Solving  Simple  Equations. 

16.  An  Axiom  is  a  general  statement  not  requiring  proof. 

Examples  of  axioms  are : 

17.  If  equal  quantities  are  ijtcreased  or  dimin- 
ished BY  THE  SAME  AMOUNT,  THEY  REMAIN  EQUAL. 

18.  If  equal  quantities  are  multiplied  or  divided 

BY  the  same  amount,   THEY  REMAIN  EQUAL. 

19.  Upon  these  hvo  axioms  is  based  the  following  rule 
for  solving  simple  equations : 

(1)  Perform  any  indicated  multiplications  and 
unite  similar  terms. 

(2)  If  there  are  any  terms  with  —  before  them, 
add  to  each  member  the  quantities  lacking. 

(3)  Subtract   from    each  member  the  smaller 
unknown  term. 

(4)  Subtract   from    each   member   the   known 
term  that  stands  beside  an  unknown  term. 

(5)  Divide  each  member  by  the  coefficient  of  x, 

20.  Where  a  quantity  may  be  separated  into  two  factors, 
one  of  these  is  called  the  coefficient  of  the  other;  but  by  the 
coefficient  of  a  term  is  generally  meant  its  numerical  factor. 

21.  Similar  terms  are  those  that  have  the  same  letters  as 
factors.     Similar  terms  may  be  united;  e.g., 

*  Reduce  all  sums  to  tlie  denomination  of  quarters. 


16  ALGEBRA. 

dx  +  4.x  =  7x 

3x  -]-  4:X  —  2x  —  6x 

3x  +  6x  +  6  +  2x  +  2  =  10x  +  7 

7X+6-2X+3-X-4r    =    4:X+4: 

Terms  not  similar  cannot  be  united;  e.g., 

3x -\- 4:y  is  not  equal  to  7y  nor  to  7i^;   it  cannot  be 
simplified. 

EXERCISE   VI. 

Perform  indicated  multiplications  : 

1.  3(x  +  5).  4.13(1-^).  7.     0(3:?:- 17). 

2.  b{x  +  4:).  6.   11(3 -4a:).         8.   8(19  +  5a:). 

8.  7{2x  -  3).  6.  8(4  +  3a:).  9.  201(10  -  71a:). 

10.   3(a:  -  5)  +  5(3  —  x). 
Unite  similar  terms : 

11.  a:  +  ^  +  7a;  +  13y  +  5  +  a:. 

12.  3a:  -  5  +  4a:  +  2  -  5a:  +  7  -  a:. 

13.  7  -  7a:  +  3  -  a:  -  11  +  9a:  +  a:  -  I. 

14.  3  +  5a:  -  8a:  -  2  +  7a;  -  10  +  a:  +  15. 

15.  x  +  y+l  +  x-y  +  l  +  x  +  y-l-x  +  y  +  1, 

16.  13a:  -  2a:  +  17  -  5a;  -  11  -  a:  +  7.a:  -  23. 

17.  1  +  a:  +  6  -  3a:  -  10  +  7a:  —  5a:  +  4. 

18.  4  +  a:  +  4a:  +  20  -  3a;  -  8  +  2x. 

19.  208  -  29a;  -  191  +  17a:  -  17  +  53a;. 

20.  1089a;  -  2001  +  x+  1987  -  787a:  -  400  +  350a:. 
Perform  indicated    multiplications  arid  unite    similar 

terms : 

21.  3(4  -x)+  l{x  —  3)  +  5a:  -  4. 

22.  5(1  -  x)  +  13(a;  -  3)  +  11a:  -  19. 

23.  a:  +  5  +  3(2a:  -  1)  +  7  +  2(1  -  5a:)  +  10a:. 

24.  3  +  4(a;  -  5)  +  3a;  +  7  +  13(4  -  3a:)  +  11. 
26.  4(a:  -  5)  +  9(3a:  -  2)  +  a:  +  4(1  -  bx), 

26.  5(7a:  -  2)  +  13  -  5a;  +  12(3  -  2a:)  +  5. 

27.  17(a:  +  5)  +  50(3  -  a:)  +  40a;  -  10  +  3(a:  +  1). 

28.  40(2a:  -  1)  +  45(2  -  a:)  +  17  +  1^(^  +  l*^)- 


THE  FIRST   USE  OF  ALOEBUA.  17 

29.  30(2:?;  +  3)  -  15^;  +  5  +  5(4  -  bx)  +  23. 

30.  I00{x  +  100)  -  10:*;  -  10  +  10(2  -  x)  -  90a:. 

Find  the  value  of  x  in  the  following  equations : 

31.  3(a;  -^l)=2x  -  11. 

32.  ^{x  +  5)  =  lOo;  -  6. 

33.  %x  -\-  b  =  ^X. 

34.  bx-1  =^  3(a;  +  1)  _  4. 
36.  lOx  +  13  =  ^x  +  9). 

36.  7(0;  -  2)  =  4t{x  +  4)  +  3. 

37.  a;  +  5  =  l{x  —  10)  —  x. 

38.  8:r  —  1  =  5^:  +  41. 

39.  1{x  +  2)  =  ic  +  32. 

40.  b{l  —  x)  =  1  —  3^;. 

41.  5(:i;  +  4)  =  14a;  +  2. 

42.  3(a;  +  3)  =  3(3a;  +  1)  -  12. 

43.  4(a;  +  5)  =  6a;  +  6. 

44.  5(a;  +  1)  =  6(3  -  a;)  -  2. 

45.  2(a;  -  3)  =  a;  +  7. 

46.  2a;  =  5(a;  —  1)  —  2a;. 

47.  17(a;  -  2)  =  2(2a;  +  1)  +  a;. 

48.  13(5  -  a;)  =  3a;  -f  1. 

49.  2a;  —  2  =  a;  +  4. 

50.  5a;  —  8  =  6(a;  —  3). 

51.  3(2a;  -  6)  =  4a;  +  1. 

52.  25  -  4a;  =  2(a;  -  4)  -  3. 

53.  7  -  a;  =  3(4  -  a;)  +  1. 

54.  2(a;  +  5)  =  5(5  -  a;)  +  a;  -  3. 

55.  4  +  a;  +  4(5  -  a;)  =  3a;  +  2(4  -  x), 

56.  2a;  +  5  =  5(a;  —  2)  +  2a;  —  10. 

57.  \+x  +  3(2  -x)  =  7(3  -  x). 

58.  13(8  -  a;)  +  2(a;  -  6)  =  2a;  +  1. 

59.  23(10  -x)  -  4(a;  -7)  =  3a;  +  3(a;  -13). 

60.  9(a;  +  1)  +  5  =  5a;  +  6(a;  -  1). 


18  ALGBBMA. 

61.  Hx  -  4)  =  4.{x  -  3)  -  17. 

62.  11(2:  +  3)  -  100  =  9(2;  +  5). 

63.  15(7^  -  3)  =  8(13^?;  +  7)  -  20. 

64.  7(19:?:  -  21)  =  11(12.-?;  -  3)  -  30. 

65.  9(252;  +  11)  =  19(12a;  ~  5)  +  2. 

66.  51a;  -  1135  +  13(103  -  x)  =z  11(2.t  +  20). 

67.  203  -  17a;  =  d{x  -  505)  -  2. 

68.  1170  +  {x-  4:)  =  ll(a;  -  3)  +  22. 

69.  9(103a;  -  820)  =  206(2  -  x)  +  139. 

70.  800(:?;  -  10)  +  2(lla;  -  32  )  -  1  -  35a;. 

71.  3000(a;  -  5)  =  10(100  -  27a;)  +  355  -  x, 

72.  10(2a;  -  3)  =  2(a;  +  15)  +  6. 

73.  3(3a;  -  11)  +  2(5a;  +  2)  =  5  +  9(2a;  -  5). 

74.  5(7a;  -  2)  +  2(5a;  -  13)  =  3(lla;  +  2)  +  26. 

75.  17(3a;  -  1)  +  3(5a;  -  17)  =  ll(5a;  -  3)  +  a;  +  5. 

76.  13(a:  -  11)  +  17(2a;  -  27)  -  1  =  3(a;  -  29). 

77.  101(5a;  +  17)  =  203(3a;  +  23)  +  3a;  -  3059. 

78.  4:{x  -  13)  =  13(a;  -  4)  -  81. 

79.  117(a;  -  25)  +  25(a;  +  117)  =  3a;  +  50  +  3(7  -  x). 

80.  8(a;  -  1)  +  4(9  -  4a;)  =  4  +  17(3  -  x). 

81.  17(2a;  +  4)  +  13(2a;  -  5)  ==  3(a;  +  17). 

82.  51a;  -  1137  +  13(103  -  x)  =  ll(2a;  +  20)  -  2. 

83.  f  (18a;  -  252)  =  2(15  _  3^)  _|_  1. 

84.  |(24a;  -  800)  =  f  (45  -  5a;)  -  5. 

85.  |(60a;  -  875)  =  ^(35  +  10a;)  -  ^x, 

86.  f  (1185a;  -  5925)  =  -^\{Ux  -  55). 

87.  i(93a;  -  138)  =  |(10a;  -  25). 

88.  4(56a;  +  42)  =  5(2a;  +  16). 

89.  f  (42a;  _  48)  -  a;  =  3(a;  +  17). 

90.  1(60  -  22a;)  +  ^^(30  -  50a;)  =  0. 

91.  i(5x  +  25)  +  |(4a;  -  28)  =  |(3a;  +  9). 

92.  |(21a;  -57)  =  2(6a;  +  17). 

93.  13a;  +  3  +  f  (18  -  12a;)  =  3(7  -  a;)  +  11  -  a;. 

94.  31(a;  -  22)  +  |(21a;  +  18)  =  2a;  +  883. 


THE  FIRST  USE  OF  ALGEBRA.  19 


Sum  of  the  Unknown  Numbers  Given. 

22.  Model  F. — A  merchant  has  grain  worth  9  cents  per 
peck^  and  other  grain  worth  13  cents  per  peck ;  in  what 
proportion  must  he  mix  40  bushels  so  that  the  mixture  may 
be  worth  40  cents  per  bushel  ? 

X  =  number  of  bushels  at  36  cents. 
40  —  a;  =  number  of  bushels  at  52  cents. 

©  3Qx  +  52(40  -x)      =  1600 

©36:^  +  2080    -522;    =1600  same  as  ® 

(D  2080   —Ux   =1600  same  as  (T) 

®  2080  =  1600  +  16a:  (3)  +  IQx 

®  480  =  lQx  ®  -  1600 

(D  30  =  :i;  ^  ®  -f-  16 

Ans.  30  bushels  of  the  cheaper  and  10  of  the  dearer. 

EXERCISE  VII. 

1.  Twenty  coins,  quarters  and  halves,  came  to  $5.75. 
Number  of  each  ? 

2.  $4.04  in  dollars,  dimes,  and  cents;  44  coins  in  all;  9 
times  as  many  dimes  as  dollars.     Number  of  each  ? 

3.  108  coins,  dimes  and  cents,  amount  to  $4.32.  Num- 
ber of  each  kind  ? 

4.  Fourteen  coins,  dimes  and  three-cent  pieces,  came  to 
77  cents.     Number  of  each  kind  ? 

5.  A  bridge  broke  down  under  a  strain  of  28,500  pounds, 
caused  by  a  crowd  of  200  people.  The  average  weight  of 
a  man  being  150  pounds,  and  of  a  woman  120  pounds,  how 
many  men  were  there  in  this  crowd  ? 

6.  Bought  30  pounds  of  sugar  of  two  sorts  for  $2.27; 
the  better  cost  10  cents  per  pound,  and  the  poorer  7  cents„ 
Number  of  pounds  of  each  sort  ? 


20  ALGEBRA, 

,._  7.  Bought  15  apples  for  59  cents;  red  ones  at  5  cents, 
green  ones  at  3  cents.     ISTumber  of  each  ? 

8.  Find  four  consecutive  numbers  whose  sum  is  94. 

9.  A  grocer  is  offered  $15  for  50  pounds  of  tea,  and 
mixes  25-cent  tea  with  35-cent  tea  so  as  to  make  $1  on  the 
transaction.     How  many  pounds  of  each  grade  of  tea  in, 
the  mixture  ? 

10.  A  merchant  has  grain  worth  11  cents  per  peck,  and 
other  grain  worth  15  cents  per  peck;  in  what  proportion 
must  he  mix  20  bushels  so  that  the  mixture  may  be  worth 
48  cents  per  bushel  ? 

11.  $1.97  in  two-cent  pieces,  three-cent  pieces,  and 
dimes ;  3  times  as  many  three-cent  pieces  as  two-cent  pieces ; 
40  coins  in  all.     Number  of  each  ? 

12.  A  man  starts  on  a  wager  to  walk  2000  miles  in  50 
days;  after  travelling  for  30  days  he  finds  that  he  can  go  5 
miles  slower  per  day  and  still  come  out  20  miles  ahead. 
How  fast  did  he  go  at  the  start  ? 

13.  One  hundred  coins,  quarters  and  dimes,  amount  to 
120.20.     Number  of  each  ? 

REVIEW. 

I.  Find  the  value  of  the  unknown  number  represented 
by  the  letter  x  in  each  of  the  following  equations  : 

1.  b{x  -l)  =  Zx  —  ^.  11.  ll(a:  +  3)  =  10{x  -f  1)  +  1. 

2.  ^x  -  3)  =^  5^  -  3.  12.  S{x  -f  10)  =  n\x  +  7)  -  8. 

3.  ^{x  -f  5)=  6:?;  -  20.  IS.  100(a;  -  5)  =  60(a;  +  3). 
^.  3(a;  +  2)  =  4:^-5.  U.  4.{x  -f  5)  =  2(3^  -  10). 

5.  6(a;- 3)  =  5(:?;-2).  15.  3(a;-10)  +  2(:?;+4)  =  6^-22. 

6.  %[x  -  ^)  =  bx  -  ^.  16.  8(10  -  x)  =z  b{x  +  3). 

7.  b(x  +  4)  =  Ix  -  4.  17.  3.T+4(:?;-2)  =  l+3(2:z;-3). 

8.  ll(a;+l)==13x-fl.  18.  l^{x-2)  =  ^^x-b). 

9.  l^x  -  10)  =:  ^x.  19.  3(3:^;  -  5)  =  4.[x  -f  5). 
10.  9(:^:+6)  =  10(22:+l).  20.  \b{bx  -  3)  =  12(4:?:  +  3). 


THE  FIRST   USE  OH  ALGEBRA.  21 

II.  Solve  the  following  problems: 

1,  Borrowed  some  coins,  and  paid  back  4  fewer  coins  of 
another  sort;  the  coins  borrowed  v,^ere  quarters,  those  re- 
paid halves;  paid  25  cents  too  much. 

2,  Borrowed  some  coins,  and  paid  back  5  fewer  coins  of 
another  sort;  the  coins  borrowed  were  quarters,  those  re- 
paid were  halves;  still  owed  50  cents. 

3,  Borrowed  some  coins,  and  paid  back  33  more  coins  of 
another  sort;  the  coins  borrowed  were  halves,  those  repaid 
were  nickels;  still  owed  $1.50. 

^.  Borrowed  some  coins,  and  paid  back  52  less  coins  of 
another  sort;  the  coins  borrowed  were  three-cent  pieces, 
those  repaid  were  quarters;  paid  $1.30  too  much. 

5,  Borrowed  some  coins,  and  paid  back  3  less  coins  of 
another  sort;  the  coins  borrowed  were  two-cent  pieces, 
those  repaid  were  dimes;  paid  74  cents  too  much. 

6.  Lost  a  few  quarters,  then  found  5  more  coins  than  I 
lost,  only  these  coins  were  the  old-fashioned  twenty-cent 
pieces;  on  the  whole  I  was  the  gainer  by  65  cents.  How 
many  quarters  did  I  lose  ? 

III.  What  is  the  difference  in  treatment  between  an 
equation  and  an  expression  ? 

IV.  What  important  principle  of  multiplication  is  illus- 
trated in  the  solution  of  the  equations  in  I  ? 

V.  Give  the  exact  meaning  of  the  word  coefficient;  also 
the  sense  in  which  it  is  generally  used. 

EXERCISE  VIII. 

Find  the  ages  of  the  jjersons  described  in  the  following 
prollems : 

1.  A  is  12  years  older  than  B;  in  2  years  he  will  be  4 
times  as  old  as  B. 

2.  A  is  6  times  as  old  as  B;  in  4  years  he  will  be  only  4 
times  as  old  as  B. 


22  ALGEBRA, 

3.  A  is  18  years  older  than  B;  in  5  years  he  will  be  3 
times  as  old  as  B. 

4.'  A  is  9  times  as  old  as  B;  in  6  years  he  will  be  only  6 
times  as  old  as  B. 

5.  A  is  7  years  younger  than  B;  in  4  years  he  will  be 
half  as  old  as  B. 

6.  In  3  years  A  will  be  \  as  old  as  B;  now  he  is  20  years 
younger  than  B. 

7.  A^s  age  at  present  is  \  of  B^s;  10  years  ago  it  was  \  of 
B's  age. 

8.  A  was  8  times  as  old  as  B  one  year  ago;  he  is  now 
only  5  times  as  old  as  B. 

9.  A  is  30  years  older  than  B ;  5  years  hence  his  age  will 
be  6  times  B's. 

10.  A  is  7  times  as  old  as  B;  in  10  years  he  will  be  only 
twice  as  old  as  B. 

The  Problem  of  the  Digits. 

23.  Model  G. — In  a  number  of  two  digits,  the  first  digit 
is  double  the  second;  and  if  27  be  subtracted  from  the 
number,  the  digits  are  reversed.     Find  the  number. 

Let  X  —  the  units  figure;  then  2^:  —  the  tens  figure. 
The  value  of  the  number  is  10  X  2^;  +  x. 
Q  20x-\-x  —  21  =  lOx  +  2:^ 
©  21x  —  27  =  12a;  same  as  © 

(D  21:^  =  12:?;  +  27  @  +  27 

©  9a;  =  27  (D  —  12:?; 

(5)  x  =  Z  ©  -^9 

Ans,   63. 

EXERCISE   IX. 

1.  In  a  number  of  two  digits  the  first  digit  is  3  times 
the  second;  and  if  36  is  subtracted  from  the  number,  the 
digits  are  reversed.     Find  the  number. 


THE  FIRST   USE  OF  ALGEBRA,  23 

2.  In  a  number  of  two  digits  the  second  number  is  3 
times  the  first;  adding  54  reverses  the  digits.  Find  the 
number. 

3.  In  a  number  of  two  digits  the  first  digit  is  4  times 
the  second;  subtracting  54  reverses  the  digits.  Find  the 
number. 

4.  In  a  number  of  two  digits  the  first  digit  is  twice  the 
second;  subtracting  36  reverses  the  digits.  Find  the 
number. 

5.  The  sum  of  the  two  digits  of  a  number  is  11;  and  if 
27  be  added  to  the  number,  the  digits  are  reversed.  Find 
the  number. 

6.  The  first  digit  of  a  number  less  than  100  is  1  less 
than  double  the  second;  and  if  18  be  taken  from  the  num- 
ber, the  digits  are  reversed.     Find  the  number. 

7.  The  sum  of  the  two  digits  of  a  number  is  9 ;  and  if  45 
be  added  to  the  number,  the  digits  are  reversed.  Find  the 
number. 

8.  The  difference  of  the  two  digits  of  a  number  is  1  less 
than  the  units  figure;  and  if  18  be  taken  from  the  number, 
the  digits  are  reversed.*     Find  the  number. 

9.  The  sum  of  the  two  digits  of  a  number  is  11;  and 
subtracting  45  reverses  the  digits.     Find  the  number. 

10.  The  smaller  digit  of  a  number  is  5  less  than  3  times 
the  larger;  and  if  9  be  added  to  the  number,  the  digits  are 
reversed.     Find  the  number. 

11.  The  difference  of  the  two  digits  of  a  number  is  1  less 
than  3  times  the  smaller;  and  if  45  be  added  to  the  number, 
the  digits  are  reversed.     Find  the  number. 

12.  The  sum  of  the  digits  of  a  number  is  11  less  than  3 
times  the  larger;  and  if  27  be  added  to  the  number,  the 
digits  are  reversed.     Find  the  number. 

*  The  fact  tliat  subtraction  reverses  the  digits  shows  which  digit 
is  the  less. 


24:  ALGEBRA. 


Numbers  in  Other  Scales. 

24.  Model  H. — The  sum  of  the  two  digits  of  a  number 
is  7;  in  the  scale  of  6  the  number  would  be  16  less.  Find 
the  number. 

Let  X  =  the  figure  in  the  tens  place;    then  "7  —  x  =^ 
the  figure  in  the  units  place. 

10x-]-7  —  X  =  the  value  of  the  number  in  the  ordinary 
or  decimal  scale; 

6x-\-7  —  X  =  the  value  of   the  number  in   the  scale 
of  6. 

©  10a;  +  7  -  :z;  ==  16  +  6a;  +  7  -  a; 
@     9x  +  7  =^x  +  23  same  as  © 

(D  4a;  =  16  @  -  7  -  5a; 

0  x  =  4:  (3)-^4 

(D  7-a;  =  3 

Ans,  43. 

EXERCISE  X. 

1.  The  first  digit  of  a  number  exceeds  the  second  by  5; 
and  if  the  number  were  in  the  scale  of  7  instead  of  the  deci- 
mal scale  of  notation,  its  value  would  be  21  less.  Find  the 
number. 

2.  The  first  digit  of  a  number  is  less  than  the  second  by 
3;  and  if  the  number  were  in  the  scale  of  12  instead  of  the 
decimal  scale  of  notation,  its  value  would  be  4  more.  Find 
the  number. 

3.  The  sum  of  the  two  digits  of  a  number  is  9;  and  if 
the  number  were  in  the  scale  of  11,  its  value  would  be  7 
more.     Find  the  number. 

4.  The  sum  of  the  two  digits  of  a  number  is  13;  and  if 
the  number  were  written  in  the  scale  of  8,  its  value  would 
be  12  less.     Find  the  number. 

5.  The  sum  of  the  two  digits  of  a  number  is    10;   if 


THE  FIRST  USE  OF  ALGEBRA.  25 

written  in  the  scale  of  8,  the  number  would  be  6   less. 
Find  the  number. 

6.  The  first  digit  of  a  number  is  twice  the  second;  if 
written  in  the  scale  of  7,  the  number  would  be  18  less. 
Find  the  number. 

7.  The  first  digit  of  a  number  is  3  times  the  second;  if 
written  in  the  scale  of  11,  the  number  would  be  9  more. 
Find  the  number. 

8.  The  first  digit  of  a  number  is  1  more  than  twice  the 
second;  if  written  in  the  scale  of  8,  the  number  would  be 
14  less.     Find  the  number. 

9.  The  second  digit  of  a  number  is  1  less  than  3  times 
the  first;  in  the  scale  of  11  the  number  would  be  3  more. 
Find  the  number. 

10.  The  sum  of  the  two  digits  of  a  number  is  1  more 
than  twice  the  second;  in  the  scale  of  13  its  value  would 
be  12  more.     Find  the  number. 

11.  The  sum  of  the  two  digits  of  a  number  is  17;  in  the 
scale  of  7  its  value  would  be  27  less.     Find  the  number. 

12.  The  first  digit  of  a  number  exceeds  the  second  by 
2 ;  if  written  in  the  scale  of  8,  the  number  would  be  14  less. 
Find  the  number. 

12.  The  sum  of  the  two  digits  of  a  number  is  7 ;  in  the 
scale  of  6  the  number  would  be  20  less.     Find  the  number. 

14.  The  two  digits  of  a  number  are  equal;  in  the  scale 
of  11  the  number  would  be  1  more.     Find  the  number. 

Current  Problems. 

25.  Model  I. — A  man  who  can  row  5  miles  per  hour  in 
still  water  finds  that  it  takes  him  5  hours  to  row  up-stream 
to  a  point  from  which  he  can  return  in  4  hours.  How  fast 
does  the  current  flow  ? 

Let  X  =  number  of  miles  per  hour  the  current  flows; 


26  ALGEBRA, 

Then  5  +  a;  =  number  of  miles  per  hour  the  man  can 

row  down-stream, 
And  b  —  X  —  number  of  miles  per  hour  the  man  can 

row  up-stream. 

®  5(5  -x)  =  4(5  +  x) 

@  25  —  5^;  =  20  +  4^;  same  as  © 

(D  25  =  20  +  9a;  ©  +  5a; 

®  5  =  9a;  ®  -  20 

®  1  =  ^  ®-9 

Ans,  Kiver  flows  |  of  a  mile  per  hour. 

EXERCISE  XI. 

1.  A  river  flows  2  miles  per  hour,  and  a  fisherman  finds 
that  he  can  row  up-stream  a  few  miles  in  6  hours,  but  it 
takes  him  only  3  hours  to  come  back.  How  fast  can  the 
fisherman  row  in  still  water  ? 

2.  A  man  starts  to  row  down  the  river  to  the  steamboat 
landing  and  back  in  4  hours;  it  takes  him  2  hours  to  get 
down  there,  and  at  the  end  of  his  time  coming  back  he  is  4 
miles  short  of  his  starting-place.  The  man  can  row  3  miles 
per  hour.  How  fast  does  the  river  flow  ?  How  far  off  is 
the  steamboat  landing  ? 

3.  I  lent  my  steam  launch  to  a  friend  who  did  not  know 
that  it  would  run  only  10  hours  without  refiring.  He  went 
down  the  river,  which  has  a  2-mile  current,  for  6  hours, 
and  on  his  return  trip  the  launch  stopped  30  miles  away 
from  the  boat-house.     What  is  the  speed  of  the  launch  ? 

4.  A  man  who  can  row  5  miles  per  hour  finds  that  it 
takes  him  3  times  as  long  to  go  up  a  river  as  to  go  down; 
find  the  speed  of  the  current. 

5.  The  two  fire-cisterns  in  a  certain  town  are  of  equal 
size,  and  are  being  filled  from  mains  that  pour  into  each 


THE  FIRST   USE  OF  ALGEBRA,  27 

10,000  gallons  per  hour;  but  the  one  near  the  business  part 
of  the  town,  being  considered  the  case  of  more  urgent  need, 
has  a  pump  rigged  from  the  other  cistern  so  as  to  hasten 
its  filling.  Consequently  it  is  half -full  in  7  hours,  while 
the  other  cistern  does  not  get  half -full  till  6  hours  later. 
What  is  the  capacity  of  the  pump  ?     Of  the  cisterns  ? 

6.  I  had  4  hours  to  spare,  and  started  for  a  row  down- 
stream ;  but  I  forgot  that  the  current  flowed  a  mile  an  hour, 
so  I  took  half  my  time  before  I  turned  to  go  back;  the 
consequence  was  that  I  got  home  two  hours  late.  How  fast 
can  I  row  in  still  water  ? 

7.  A  man  can  row  4  miles  per  hour  in  still  water;  going 
up-stream  he  goes  only  f  as  fast  as  he  comes  down.  Find 
the  rate  of  the  stream. 

8.  A  man  who  can  row  6  miles  per  hour  in  still  water 
goes  1  mile  farther  in  3  hours  coming  down-stream  than  he 
does  in  4  hours  going  up.     Find  the  speed  of  the  current. 

9.  I  had  6  hours  to  spare,  and  started  for  a  row  down- 
stream ;  but  I  forgot  that  the  current  flowed  1 J  miles  per 
hour,  so  I  took  half  my  time  before  I  turned  to  go  back; 
the  consequence  was  that  I  was  only  half-way  home  when 
my  time  was  up.     How  fast  can  I  row  in  still  water  ? 

10.  A  centipede  in  a  grain- elevator,  crawling  on  a  belt 
which  moves  11  feet  per  second,  takes  10  seconds  to  go  the 
whole  length  of  the  building  when  he  goes  with  the  belt, 
and  2  minutes  to  return,  against  the  motion  of  the  belt. 
How  fast  could  the  centipede  go  if  the  belt  were  still  ? 

11.  A  flsh  that  can  swim  15  miles  per  hour  takes  12 
minutes  to  pass  up  through  a  certain  part  of  the  river,  but 
takes  only  10  minutes  to  come  down  again.  Speed  of  the 
current  there  ? 


ALGEBRA. 


New  Work  on  Old  Patterns. 

26.  In  Problem  I  mark  a  small  letter  a  over  the  number 
5  in  "  row  5  miles/'  a  letter  h  over  the  number  5  in  ^^  takes 
him  5  hours/^  and  a  letter  c  over  the  number  4  in  *'  return 
in  4  hours."  Then  solve  the  problem,  putting  instead  of" 
the  numbers  marked  a,  b,  and  c,  the  values  represented 
below,  each  set  making  a  new  problem. 


a 

b 

c 

1. 

4 

3 

1 

2. 

6 

4 

2 

3. 

H 

8 

3 

4, 

^ 

4 

2 

5. 

H 

4 

3 

In  example  1  of  the  preceding  exercise,  letter  the  numbers 
2,  6,  and  3  in  the  same  way,  and  then  use  the  following 
sets  of  values  instead,*  just  as  in  the  case  of  Problem  I: 


a 

h 

c 

1, 

li 

5 

2 

2. 

1 

6 

5 

3. 

3 

8 

5 

4. 

3 

4 

1 

5. 

n 

8 

3 

FRACTIONAL   EQUATIONS. 

27.  When  equations  arise  that  contain  fractions,  it  is 
necessary  to  multiply  both  sides  of  the  equation  by  num- 

*  For  some  of  the  problems  suggested  here   it  may  prove  more 
reasonable  to  describe  the  powers  of  a  fish  instead  of  a  fisherman. 


THE  FIRST  USE  OF  ALGEBRA,  29 

bers  such  as  will  change  the  fractions  to  whole  numbers,  as 
in  the  following  examples  : 


®  X  3 

®  X  3 
©  X  5 
(D-IO 

One  line  of  the  work  in  example  K  could  have  been  saved 
if  we  had  multiplied  both  sides  of  (\)  by  15  at  once,  instead 
of  first  by  3  and  then  by  5. 

Model  L.— (T)      -  _  -  r=  -  -  i 


Model  J.- 

-®i 

■  =% 

©   X 

=  6 

Model  K. 

-® 

3 

3 

''  5 

® 

2y  = 

9 
5 

®     y  = 

:  9 

9 
10 

^-.                 2x        X         . 

®  X  2 

@  3a;  -  2a;  =  y  -  3 

©  X  3 

®              «=|-3 

same  as  © 

(D             2a;  =  3a;  —  6 
®  2a;  4-  6    =  3a; 
®              6  =  a; 

®  X  2 
® +  6 
®  -  2a; 

Here  if  we  had  multiplied  Q  by  12,  the  least  common 
multiple  of  the  denominators,  we  should  have  had; 

Model!.— (T)       ---  =  --1 

'^       2       3       4      2 

®  ^x-A:X^^x-Q  ®  X  12 

(D  2x  —  '6x  —  ^  same  as  @ 

®  2a; +6=  3a;  (D  +  6 

®  6  =  ;2;  ®  -  2a; 


30  ALGEBRA, 


EXERCISE  XII. 

Solve  the  following  equations  {that  is,  find  the  value  of  x 
implied  hy  them)  : 

1.   |+|+3=«:-4.       6.  1-^^+17  =  0. 

a;a;l_  2       a;_5a;  —  8 

^-  2  +  3+3  ~^~'-^-       ''•   ~3  +  6  =  ~36~* 

X      ^_^ii  2a;  —  1       1_K    I  a; 

^-  3  ~  7  ~  6  +  ^-  ■  '•  ~%  3  ~  ^  +6- 

XX      1  _x  -{-  2  7a;— 6      x      3a;  —  5 

*-4  +  6~3-~3~-       ^-  ~T~  +  3"=~~3~  +  ^~^• 
a;      a;_a;  3a;  —  7       3a;      x  —  b 

'*-2~5-2~^-         ^*'- ^        n  =  ^~- 

28.  Model  M. — Bought  a  certain  number  of  picture- 
cards  at  2  for  a  nickel,  and  the  same  number  at  3  for  a 
nickel.  Sold  them  all  at  a  uniform  price  of  2  cents  apiece, 
and  lost  a  quarter.     How  many  did  I  buy  ? 

Let  X  =  the  number  of  cards  of  each  kind;  then  — 

2 

would  be  the  cost  of  one  lot  and  —  of  the  other. 

o 

©       |  +  |  =  4a;  +  25 

(2)     6x  +  —  =  8x  +  60  ©  X  2 

(D     5x  +  ^0x  =  24:x+160  ©X3 

(4)  25.'?;  =  2ix  +  150  same  as  @ 

®  a;  =  150  ®  -  Ux 


THE  FinST   USE  OF  ALGEBRA,  81 

EXERCISE  XIII. 

1.  What  number  is  that  |  of  which  exceeds  27  by  as 
much  as  ^^  of  it  is  less  than  6  ? 

2.  A,  B,  and  C  shared  a  sum  of  money  so  that  A  had 
$30  more  than  half,  B  $33  more  than  a  fourth,  and  0  the 
remainder,  which  was  $16.     Find  the  shares  of  A  and  B. 

3.  A  has  $107,  B  $45.  B  gives  A  a  certain  sum  and 
then  has  only  ^  as  much  as  he.     How  much  did  B  give  A  ? 

4.  If  a  certain  number  be  successively  subtracted  from 
53  and  62,  and  then  ^  of  the  first  remainder  be  taken  from 
4  of  the  second,  the  last  remainder  will  be  10.  Find  the 
number  first  subtracted. 

5.  A  and  B  started  eastward  from  Palmer.  At  noon  A 
had  gone  f  of  the  distance  to  Boston,  B  had  gone  ^  of  it, 
and  they  were  just  2  miles  apart.  What  was  the  whole 
distance  to  Boston  ? 

6.  From  a  cask  of  wine  -^^  had  leaked  out,  f  had  been 
drawn  out,  and  there  were  11  gallons  left.  How  much 
was  there  at  first  ? 

7.  Bought  pineapples  at  the  rate  of  $4  for  7  dozen,  and 
sold  them  at  the  rate  of  $2  for  3  dozen,  gaining  $4.  How 
many  pineapples  were  there  ? 

8.  A  man  can  walk  4  miles  an  hour  or  ride  7  miles  an 
hour.  How  long  will  it  take  him  to  go  66  miles,  if  he  rides 
half  the  time  and  walks  half  the  time  ? 

9.  The  fore  wheel  of  a  carriage  has  a  circumference  of  6 
feet,  and  the  hind  wheel  a  circumference  of  10  feet.  When 
the  fore  wheel  has  made  25  turns  more  than  the  hind  wheel, 
how  far  has  the  carriage  gone  ? 

10.  A  woman  had  a  basket  of  eggs,  and  sold  -J  of  them, 
with  i  of  an  Qgg  besides;  what  she  had  left  were  3  more 
than  half  what  she  had  at  first.  How  many  eggs  had  she 
at  first  ? 

11.  A  boy  has  an  hour  for  exercise;  how  far  may  he  ride 


32  ALGEBRA, 

with  his  father,  at  the  rate  of  10  miles  per  hour,  before 
getting  out  to  walk  back  ?     He  can  walk  3-J  miles  per  hour. 

REVIEW. 

1.  What  is  an  axiom  ?  Upon  what  axiom  does  the  re- 
duction of  fractional  equations  depend  ? 

II.  What  would  express  the  value  of  a  number  whose 
three  figures  were  represented  by  x^  y,  and  z  ?  What  con- 
clusion could  you  draw  as  to  the  relative  value  of  these 
letters,  if  you  were  told  that  subtracting  198  reverses  the 
order  of  the  figures  ? 

III.  Solve  these  problems: 

i.  Paid  $3.75  with  halves  and  quarters;  3  times  as  many 
quarters  as  halves.     Number  of  each  ? 

2.  Paid  $2.45  with  dimes  and  nickels;  5  times  as  many 
nickels  as  dimes.     Number  of  each  ? 

3.  Paid  $1.36  with  three-cent  pieces  and  two-cent  pieces; 
7  times  as  many  2's  as  3's.     Number  of  each  ? 

Jf..  Paid  $1  with  cents  and  two-cent  pieces;  18  times  as 
many  cents  as  two-cent  pieces.     Number  of  each  ? 

5,  Paid  $2.26  with  dimes,  two-cent  pieces,  and  halves;  3 
more  two-cent  pieces  than  dimes;  25  coins  in  all.  Number 
of  each  ? 

6.  Paid  $7.50  with  nickels,  quarters,  and  halves;  1  more 
quarter  than  halves;  55  coins  in  all.     Number  of  each  ? 

7.  Paid  $3.23  with  two-cent  pieces,  three-cent  pieces, 
and  dimes;  14  more  dimes  than  two-cent  pieces;  49  coins 
in  all.     Number  of  each  ? 

8,  Paid  $5.90  with  three-cent  pieces,  dimes,  and  quar- 
ters; 6  fewer  dimes  than  quarters;  85  coins  in  all.  Number 
of  each  ? 

P.  Paid  $2.24  with  halves,  two-cent  pieces,  and  nickels; 
5  more  nickels  than  halves;  28  coins  in  all.  Number  of 
each? 


THB  FIB8T  USE  OF  ALGEBRA,  33 

10,   Paid  $2.66  with  cents,  quarters,  and  dimes;  2  more 
dimes  than  cents;  29  coins  in  all.     Number  of  each  ? 

EXERCISE  XIV. 

Find  the  value  of  x  in  the  following  equations: 

X  —  1      X       _  ,.  2x+  1      4:X       ^ 

1.   —, =  2a;  -  10.         4.   — ^ =2  -X, 

4  5  7  6 

X  4-  1   ,   X  4-  3  ^  X  4-  5   ,   X  —  5       dx 

a:-2       3-x    ^    ^ 

7.  3x  -\ ^  =  — ^ h  ^^. 

x-l       32;-  1       1  _  ix-  3       g;-  7 

8.  r^+       5        +7-        5        +3* 

9.  32;   -f  5   H ^r—   = h  52;. 


32;  —  1    ,    15^;  —  3    ,    ^ 
10.   IO2;  = ^ 1 ^^  4-7^-1. 

The  Minus  Sign  before  a  Fraction. 

29.  In  example  10  of  the  preceding  exercise  the  equation 

would  have  assumed  quite  a  different  form  if,  say,  the  frac- 

]^5^ 3 

tion had  been  subtracted  from  each  side  of  the 

equation. 

It  would  then  be 

^  in         15^;- 3       32;-l    ,    ^ 

Q  10^' ^—  =  — ^ h  ^^  -  1, 

and  in  solving  this  especial  pains  must  be  taken  on  account 
of  the   fact    that   when  we   multiply  by  14  the  fraction 


34  ALGEBRA. 

jr becomes  a  whole  number^  and  that  whole  number 

has  to  be  subtracted  from  the  preceding  term  of  the  equa- 
tion. 

(2)  140a;  -  (30a;  -  6)  =  21a;  -  7  +  98a;  -  14. 

Here  we  have  30a;  —  6  to  subtract  from  140a;. 

If  we  had  to  take  30a;  from  140a;,  the  remainder  would 
be  110a;;  now,  however,  since  we  are  taking  6  less  than 
30a;,  our  remainder  will  be  6  more,  or  llOo;  -|-  6. 

The  equation  is  therefore  solved  as  follows : 

Q  1 0:.  _  1-^^  ^  ^-i  +  7 a;  -  1 

©  140a;  -  (30a;  -  6)  =  21a;  -  7  +  98a;  -  14  ®  X  14 

(D  110a; +6    =  119a;  —  21  same  as  @ 

©  110a;  +  27  =  119a;  @  +  21 

®  27  =  9a;  ©  -  llOo; 


EXERCISE  XV. 

1. 

o;  -  1      a;  +  3      2  . 
2              5      ~5' 

..  +  ^  +  '-3      '"  +  '-. 

2. 

5a;  -  4      3a;  +  2      a;  -  4 

8                7       ~"      4    • 

X       X  —  b       a;  —  8 

''^    4           3      -      4     • 

3. 

2a;-la;-5       a; +3 
3         '       6      ~      2     * 

_a;  —  4  X  —  b  x-\-  5 
'•2               8      ~      7     ■ 

4. 

5^-     ^^         8^+      5    . 

x  +  5  x-\-l  x-\-i 
*•       6               8      ~      4     ■ 

20  -  3a;       10 

—  3a;      X 

''         6    - 

3        ~2" 

,n    -  +  ^        ^-  + 

1  _  7a;  -  11 

15  25  45 


TBE  FIBST   USE  OF  ALGEBRA.  35 

11.  bX'\'20-  2{x  +  2)  =  40  -  4.{x  —  15). 

12.  10(a;  +  1)  -  {Sx  +  5)  =  30  -  lOo;  +  2{x  +  35). 

13.  30(a;  +  6)  -  10(a;  +  3)  =  311  -  3x, 

14.  2a;  +  3{x  +  9)  =  200  -  4(90  +  Sx), 

15.  5(a;  -  6)  +  17  =  110  -  2{x  -  5). 

16.  U{x  -  13)  +  10  =  200  -  2(75  -  x). 

17.  200  -  7(60  -  5.^)  =  3{2x  -  11)  -  100. 

18.  ^^x  —  10(a:  +  10)  =  40  +  :c  -  8(2  -  x). 

19.  274(2:?;  -  10)  -  163(8  +  ^)  -  48^;  =^  0. 

20.  ^x  -  103(a;  -  17)  +  17(3:?;  -  103)  +  135  =  0. 

21.  -V  -  100(13:^  -  74)  =  7(7:c  -  2). 

22.  I  -  \\{^x  -  25)  =  \(hx  -  61). 

23.  2.^  —  7  —  — ^r —  =  17  — 


24.   25 


2      ~^'  4      • 

a;  +  17_7.-r  — 2      8:r-  13 
2       ~       3       +        9      * 


10:?;  -  7    ,    ,  ^       62;  +  33       3:r  -  11 

25.   ^—  +  13  =  ^-  +  --^— . 

4a;  -  7       ^       3a;  -  11       a;  +  17 

26.  — ^ ^=—j ^. 

5(0; +1)       /         2a; --7\  ,^ 

lla;  +  19       6a;  —  5  _  6a;  +  1 
^®-  6  3"""       6~^' 

29. 


31. 


.,N        /         X  —  1\       X  4-1 
7a;  -  11       6a;  +  5      9a;  -  5 


4  5  22 


2a;  +  7       3a;  +  4 

a;  +  3       2a;  -  13       x       ^ 

33-  -5 rr"  =  3~^- 


36  ALGEBRA. 

Day's  Work  and  Cistern  Problems. 

30.  Model  N. — A  can  do  work  in  4  days  which  B  would 
take  12  for.     How  long  for  both  ? 

Let  X  =  the  number  of  days  for  both  to  do  the  work. 

Then  in  one  day  both  can  do  -  of  the  work. 

A  can  do  J  of  the  work. 
B  can  do  -^^^  of  the  work. 

®  i  +  r2  =  x 

®     i  +  f2^"^  ®^'' 

(D  3:^  +  a;    =  12  ©  X  12 

0  4^  =  12  same  as  (3) 

®  x  =  d  ®  -  4 

A?is.   Three  days  for  both. 

EXERCISE   XVI. 

1.  A  can  do  work  in  15  days,  B  the  same  in  18  days. 
How  long  for  both  ? 

2.  A  vessel  can  be  emptied  by  three  taps;  by  the  first 
alone  in  80  minutes,  by  the  second  alone  in  200  minutes, 
and  by  the  third  alone  in  5  hours.  How  long  if  all  three 
are  open  ? 

3.  A  can  do  a  job  in  2J^  hours  which  B  can  do  in  If 
hours  and  C  in  3J  hours.     How  long  for  all  together  ? 

4.  A  does  I  of  a  piece  of  work  in  10  days;  then  he  calls 
in  B  to  help  him,  and  they  finish  the  work  in  3  days. 
How  long  would  B  take  to  do  the  work  by  himself  ? 

5.  A  bath-tub  is  filled  in  40  minutes  and  emptied  by  the 
waste-pipe  in  1  hour.  How  long  will  it  take  to  fill  it  with 
the  waste-pipe  open  ? 


THE  FIRST   USE  OF  ALGEBRA.  37 

6.  A  cistern  can  be  filled  in  12  minutes  by  one  pipe 
alone,  or  in  8  minutes  if  the  second  pipe  is  also  turned  on. 
How  long  to  fill  it  with  the  second  pipe  alone  ? 

7.  A  cistern  holding  2400  gallons  can  be  filled  in  15 
minutes  by  three  pipes,  one  of  which  lets  in  10  gallons  more, 
and  the  other  4  gallons  less,  per  minute,  than  the  third. 
How  many  gallons  flow  through  each  pipe  per  minute  ?  * 

8.  A  can  do  a  job  by  himself  in  6  days,  B  can  do  it  in 
10  days,  and  both  together  with  C  to  help  them  in  2f  days. 
How  long  for  C  to  do  it  alone  ? 

9.  A  cistern  can  be  filled  from  the  water-main  in  12 
hours ;  if  also  an  extra  pump  is  rigged  from  a  neighboring 
cistern,  it  can  be  filled  in  8  hours;  and  with  all  the  fire- 
engines  in  town  to  help  by  pumping  water  from  a  lake 
near  by  it  could  be  filled  in  6  hours.  How  long  would  it 
take  to  fill  the  cistern  if  the  main  were  shut  off,  and  the 
filling  depended  entirely  on  the  extra  pump  ?  How  long 
if  the  filling  depended  entirely  on  the  fire-engines  ? 

10.  A,  B,  and  C  are  working  together  on  a  job  that  any 
one  of  them  could  do  alone  in  36  days;  but  when  the  job 
is  half  done  C  goes  on  a  strike,  and,  by  annoying  A  and  B, 
hinders  the  work  as  much  as  he  could  have  helped  it  if  he 
had  continued  at  work.  When  the  job  is  done  how  many 
days  must  A  be  paid  for  ?  How  long  did  0  work  ?  How 
many  days  did  his  annoyance  delay  the  completion  of  the 
job? 

11.  Starting  at  noon,  I  can  get  to  town  on  a  wagon  at 
1 :48  o'clock;  if  I  walked,  I  could  get  there  at  2 :15  o'clock. 
If  I  agree  to  walk  half  the  time,  what  time  must  I  get  off 
the  wagon  ? 

♦  The  method  of  solving  this  example  is  quite  different  from  the 
preceding  six.  It  is  introduced  here  lest  the  pupil  should  conclude 
tbat  all  '*  cistern"  examples  had  to  be  done  on  the  same  plan  as 
Model  N, 


38  ALGEBRA. 


The  Clock  Problem. 

31.  Model  0. — At  what  time  between  4  and  5  o'clock 
are  the  hands  of  a  clock  9  minutes  apart  ? 
Let  X  =  the  number  of  minutes  past  4. 

Then  —  =  the  number  of  minute-spaces  traversed  by 

the  hour-hand  in  x  minutes. 

If  the  hour-hand  is  9  minutes  ahead  of  the  minute- 
hand,  the  equation  becomes 

©30  +  ^  =  ^  +  9 

©  240  +  a;  =  12a;  +  108         ®  X  12 

(D  132  =  11^  (2)-x-  100 

®     12  =  2;  (D  -^  11 

Ans.  12  minutes  past  4. 

If  the  minute-hand  is  ahead  of  the  hour-hand,  the 
equation  becomes 

9 

-  108         ®  X  12 

©  +  108  -  a; 

(D-11 
Ans.  28/y  minutes  of  5. 

Of  these  two  answers  the  first  only  can  be  realized  on  an 
ordinary  clock,  because  the  mechanism  of  the  clock  is  such 
that  28y*y  minutes  of  5  is  never  indicated  by  the  position 
of  the  hands.  They  move  by  jerks,  one  for  each  tick  of 
the  pendulum,  and  pendulums  that  tick  elevenths  of  a 
second  would  not  have  any  reason  for  existence  other  than 
the  need  of  illustrating  this  problem. 


© 

20 

+ 

12  "" 

-  X  ' 

© 

•340 

+ 

X  = 

V%x 

(D 

348 

= 

\\x 

® 

31t\ 

= 

X 

TUB  FIEST   USE  OF  ALGEBRA,  39 


EXERCISE   XVII. 

Instead  of  the  numbers  4,  5,  mid  9,  in  the  statement  of 
Model  0,  use  the  following  sets  of  numbers,  and  solve  the 
resulting  proble7ns  : 


1.  4;  5;  2.  8.  8 

2.  3;  4;  4.  9.  8 

3.  6;  7;  3.  10.  5 


9;  7.         15.  10;  11;  21. 
9;  4.         16.  12;  1;  22. 
6;  3.  17.  4;  5;  24. 

4.  7;  8;  2.         n.  12;  1;  27.     18.  7;  8;  24. 

5.  3;  4;  7.         12.  9;  10;  26.     19.  9;  10;  23. 
6    5;  6;  14.       13.  1;  2;  21.       20.  1;  2;  17. 
7.  6;  7;  14.       u.  2;  3;  26. 

21.  A  student  of  music  always  practised  at  the  piano 
between  5  p.m.  and  7  p.m.  One  day  he  noticed  as  he 
began  to  practise  that  the  hands  of  his  watch  were  exactly 
3  minutes  apart;  he  practised  until  they  were  again  3  min- 
utes apart,  and  then  stopped.     How  long  was  he  practising  ? 

22.  In  a  mile  race  on  an  oval  track,  11  laps  to  the  mile, 
a  runner  has  200  feet  the  start  of  a  bicyclist  and  only 
moves  ^  as  fast.  Where  will  the  bicyclist  pass  the  runner 
the  first  time  ? 

23.  Where  will  the  bicyclist  pass  the  runner  the  second 
time  ? 

24.  If  they  both  start  at  the  starting-post,  and  go  oppo- 
site ways,  where  will  they  meet  the  second  time  ? 

25.  If  the  bicyclist  starts  from  the  starting-post  and  the 
runner  starts  from  the  half-way  post,  and  they  go  opposite 
ways,  where  will  they  meet  the  second  time  ? 

26.  One  boy  can  run  5  yards  while  another  runs  7;  if 
both  start  opposite  ways,  from  the  same  corner,  to  run 
around  a  block  75  yards  on  one  street  and  42  yards  on  the 
other,  how  far  from  the  furthest  corner  will  they  meet  ? 

27.  The  hands  of  a  clock  are  together  ten  different 
times  between  midnight  and  noon.     Find  when. 


40  ALGEBRA. 

28.  The  hands  of  a  clock  are  at  right  angles  twenty-two 
different  times  between  midnight  and  noon.     Find  when. 

29.  The  hands  of  a  clock  point  opposite  ways  eleven 
different  times  between  midnight  and  noon.     Find  when. 

30.  The  hands  of  a  clock  make  with  each  other  an  angle  of 
30°  twenty-two  different  times  between  midnight  and  noon. 
Find  when. 

31.  Find  when  on  an  actual  clock  the  hands  are  exactly 
one  minute-space  apart.  At  what  times  are  the  conditions 
of  the  last  four  problems  realized  on  an  actual  clock  ? 


CHAPTEK  n. 

ABBREVIATION   OF   RULES. 

32.  The  second  important  use  of  Algebra  is  for  abbre- 
viating rules  in  Arithmetic. 

For  example,  the  rule  for  finding  the  number  of  square 
inches  in  a  circle,  when  its  radius  is  known,  is:  *'  Square 
the  radius,  and  multiply  by  3.1416.''^  By  Algebra  we 
abbreviate  this  rule  to 

where  r  is  understood  to  mean  the  radius  of  the  circle  and 
71  stands  for  the  number  3.1416  (or  34-  as  it  is  sometimes 
written).*  Such  an  abbreviation  of  a  rule  is  called  a 
formula. 

Translating  Formulae  into  Rules. 

33.  Model  A. — Express  as  a  rule  the  formula 
a  —  V{h  -\-  I)){h  —  h),  which  enables  us  to  determine  the 
length,  «,  of  one  leg  of  a  right  triangle  when  the  hypote- 
nuse, h,  and  the  other  leg,  Z>,  are  given. 

Here  ^  +  ^  is  the  sum  of  the  hypotenuse  and  given  leg, 
and  A  —  ^  is  their  difference;  so,  to  find  the  required  leg, — 

Multiply  the  sum  of  the  given  sides  by  their  difference 
and  find  the  square  root  of  the  product. 

*  The  exact  value  of  the  famous  number  represented  by  7t  cannot 
be  expressed  in  figures.  To  ten  places  of  decimals  its  value  is 
3.1415926535.  The  value  3^  is  near  enough  for  ordinary  use,  and 
3.1416  is  more  accurate  still,  being  correct  to  within  j^nrirnF- 

41 


42  ALGEBRA. 

EXERCISE  XVIII. 

Translate  into  rules  the  following  formiilce : 

1.  For  the  circumference  of  a  circle,  when  the  length  of 
the  radius  is  given :  %nr, 

2.  For  the  area  of  an  equilateral  triangle,   when  the 

length  of  one  side  is  given :  — — — . 

3.  For  the  volume  of  a  circular  pillar,  when  the  radius 
and  lieight  are  given :  nrVi. 

4.  For  the  volume  of  a  square  pyramid,  when  the  height 

and  one  side  of  the  base  are  given :   — . 

o 

5.  For  the  volume  of  a  sphere,  when  the  radius  is  given : 
4;rr^ 

~~3~' 

6.  For  the  diagonal  of  a  rectangle,  when  the  length  and 
breadth  are  given :    ^V"  -\-  V^. 

7.  For  the  average  diameter  of  a  tree,  when  the  girth  is 

known:  — . 
n 

8.  For  the  diameter  of  a  ball,  when  the  volume  of  it  is 
known:    \  — . 

^       7t 

9.  For  the  number  of  crossings  made  by  a  number  of 
straight  lines  drawn  at  random:  — '^ — -.     (n  stands  for 

the  number  of  lines.) 

10.  For  the  number  of  seconds  required  for  a  body  to 

/25 

fall:  y   — .     {s  stands  for  the  distance  in  feet,  and ^  stands 

for  the  number  32.2.) 

34.   In  writing  a  formula,  numbers  that  are  given  in  the 
rule  are  written  in  figures,  except  when  the  numbers  are  of 


ABBREVIATION  OF  RULES.  43 

very  frequent  use  and  very  well  known  value.     Such  are 
the  numbers  n  =  3.1416,  g  =  32.2,  and  a  few  others. 

Translating  Rules  into  Formulae. 

35.  Model  B. — Give  the  formula  for  the  following  rule: 
To  find  the  number  of  gallons  a  pail  will  hold :  Find  the 
diameter  of  the  bottom  and  of  the  top,  and  find  the  depth 
of   the  pail;  then  multiply   the  two  diameters   together, 
square   each  diameter,    and  add  the   three  results;    then 
multiply  by  11  times  the  height  and  divide  by  10,000. 
Let  D  =  the  diameter  of  the  top  of  the  pail ; 
d  =  the  diameter  of  the  bottom  of  the  pail; 
h  =  the  height  of  the  pail. 
Multiply  the  two  diameters,  Dd  ;  square  each,  D^  and  d^ ; 
add  the  three  results,  I)^  -{-  d^  -\-  Dd  ;  multiply  by  11  times 

the  height  and  divide  by  10,000,  ^^J.D'^  +  ^^  +  Dd). 

EXERCISE  XIX. 

Oive  the  formulcB  for  the  folloiving  rules : 

1.  To  find  the  area  of  an  ellipse:  Multiply  half  the 
greatest  length  by  half  the  greatest  breadth  and  multiply 
the  product  by  3.1416. 

2.  To  find  the  tonnage  of  a  vessel  multiply  the  length 
of  the  keel  by  the  breadth  of  the  main  beam,  and  by  the 
depth  of  the  hold  in  feet,  and  divide  by  95. 

3.  To  find  the  volume  of  a  coifee-pot,  multiply  the 
greatest  and  least  radii  together;  add  to  this  product  the 
square  of  the  greatest  radius  and  the  square  of  the  least 
radius;  multiply  the  result  by  one- third  the  height;  and 
finally  multiply  by  3.1416. 

4.  To  gauge  a  cask,  take  the  difference  between  the 
head  and  bung  diameters,  multiply  by  .62,  and  add  the 


44  ALGEBRA. 

head  diameter;  square  this,  multiply  by  the  length  of  the 
cask  and  by  \  of  3.1416. 

5.  To  find  the  area  of  any  triangle  when  the  three  sides 
are  given:  Add  the  three  sides  and  divide  by  2;  subtract 
from  this  each  side  successively;  multiply  the  four  results 
together  and  find  the  square  root  of  the  product.* 

6.  To  find  the  diameter  of  a  dome  when  its  surface  is 
given,  divide  double  the  surface  by  3.1416  and  find  the 
square  root  of  the  quotient. 

7.  To  find  the  diameter  of  a  cylinder  when  its  height 
and  volume  are  given,  multiply  the  height  by  3.1416  and 
divide  four  times  the  volume  by  the  product;  then  find  the 
square  root  of  this  result. 

8.  To  find  the  depth  of  a  conical  hole  when  its  cubical 
contents  and  its  diameter  are  given,  multiply  the  square  of 
the  diameter  by  3.1416  and  divide  12  times  the  volume  by 
the  product. 

9.  To  find  the  velocity  in  feet  per  second  of  water  flow- 
ing from  a  pipe,  when  the  discharge  is  given  in  gallons  per 
minute,  and  the  diameter  is  given  in  inches,  multiply  the 
square  of  the  diameter  by  2.448  and  divide  the  discharge 
by  the  product. 

10.  To  find  the  velocity  of  water  pouring  from  a  hole 
in  the  side  of  a  tank  at  a  given  depth  (in  feet)  below  the 
surface,  multiply  twice  the  depth  by  the  number  32.2  and 
find  the  square  root  of  the  product. 

Application  of  Rules. 

36.  Model  C. — How  many  square  feet  of  flooring  are  re- 
quired to  cover  a  circular  platform  100  feet  in  diameter  ? 

Here    r  =  50,    r^  =  2500,    and 

Ttr^  =  2500  X  3.1416  =  7854  square  feet.    Ans. 
*  Let  s  stand  for  one-half  of  the  sum  of  the  three  sides. 


ABBEEVIATION  OF  RULES,  45 

EXERCISE  XX. 

1.  A  large  leaden  ball  was  dropped  into  a  tank  com- 
pletely full  of  water,  and  33510.4  cubic  centimetres  of 
water  overflowed.     What  was  the  diameter  of  the  ball  ? 

2.  A  circular  floor  contains  1256.64  square  feet.  What 
is  its  diameter  ? 

3.  It  takes  1413.72  square  yards  of  roofing- tin  to  cover 
a  hemispherical  dome.    What  is  the  diameter  of  the  dome  ? 

4.  A  cylinder  9  feet  high  and  containing  763.4088  cubic 
yards  is  how  wide  ? 

5.  A  conical  hole  in  the  ground  is  intended  to  contain 
261.8  cubic  feet  of  water,  and  it  is  10  feet  across.  How 
deep  must  it  be  in  the  centre  ? 

6.  Out  of  a  lot  of  logs  12  feet  long  I  want  to  pick  out 
those  not  having  less  than  37.6992  cubic  feet  of  lumber. 
What  must  be  the  least  possible  distance  through  ? 

7.  I  find  that  the  average  circumference  of  a  lot  of  logs 
is  9.4248  feet.  Suppose  I  cut  the  logs  up  into  chopping- 
blocks,  how  wide  would  they  be  ? 

8.  A  horse  attached  to  the  arm  of  a  windlass  walks  10.56 
feet  short  of  5  miles  in  making  300  turns.  How  long  is 
the  arm  of  the  windlass  ? 

9.  A  pipe  5  inches  in  diameter  is  to  deliver  102  gallons 
of  water  per  minute.  What  must  the  velocity  of  the  water 
be,  in  feet  per  second  ? 

10.  How  fast  will  water  flow  through  a  hole  in  a  dam 
32f  feet  below  the  surface  of  the  water  in  the  reservoir  ? 

11.  A  milk-pail  a  foot  deep  is  a  foot  across  the  top  and 
10  inches  at  the  bottom.     How  many  quarts  will  it  hold  ? 

12.  A  cask  30  inches  long  is  18  inches  across  at  the  head 
and  20  inches  through  at  the  bung.  How  many  gallons 
will  it  hold  ? 

13.  How  many  gallons  per  100  feet  of  length  can  be  con- 
tained in  a  water-main  20  inches  in  diameter  ? 


46 


ALGEBRA. 


Application  of  Formulae 

37.  Model  D.- — State  the  operations  involved  in  the  fol- 
lowing formula  and  calculate  its  value  for  the  given  values 
of  the  letters : 


W 


12     ' 


L  =  150;         s  =  8; 


W=S80, 


Statement  of  Operation, — Multiply  the  number  L  by  it- 
self; multiply  the  number  s  by  itself;  add  the  two  results, 
divide  by  12,  and  multiply  by  the  number  W. 

Evaluatio7i, 

L2  =  22500 

s^=        64 

12)22564 


W 


1880i 

380 

7145261 


EXERCISE   XXI. 

State  the  operations  involved  in  the  folloioing  formulcB,  and 
calculate  their  values  for  the  give?i  values  of  the  letters  : 


1    W- 

^'       12 

2.  Wr2 

,L2  +  s2 


3.  W- 

4.  W 

5.  W 

6.  W 


12 

t 

2 

2 
21-2 


W^SOO 
W=  210 
W  =  200 

W=S50; 

W=:JfOO; 

W=80; 


L  =  20. 
r  =  5. 

L^25] 

r  —  2. 
r  =  20; 
r=  15. 


s  =  2. 


R=z25. 


ABBREVIATION  OF  MULES.  47 

7.  w(l'  +  I')  W=  20;       L  =  10;      r  =  2. 

T  —  t 

8.  IQ— Tjr-  T=Jf59)   t=m]   Q=161100;  1=^15. 

9.  ^  F=  ^^^^;  '  ^  =  SOO;     V  =  400. 

10.  C-j-  0=100;     B  =  S;    D=10;     L  =  15. 

4LVd 

11. L=  1500;  n=1700;  d=7.6S;  g=981. 

12-^^  P  =  10;      P=12. 

mr 

13.   7  m  =  i.^;      r  =  ^^ 

m  —  1  ' 

14.  ^ —  V=-  ^\        P  =  12. 

d 

16.  T—^ ^   =  -^^^         d  =    17. 

1  +  n 
Rr 

^e-   (R+r)(m-l)  '^=^^  ^^^-^^     ^  =  ^^- 

17.  2dnm  c^  =  i7^^^;     n  =  lUO;     m  =  12.6. 

18.  n(4L  +  d)  (1=5;  L  =  30;         n  =  255. 

^^-  ^  ~  87*  30  -^ ""  '^^^'         ^  =  ^-^--^^       ^  =  ^^• 

^^-  ^  ~  ^-"^  -^  =  •^^^'        ^  =  i5;  A  =  ^4. 

360L 

21.  L  =  40;  c  =  900. 


c 

7ra 
180 
180u 

7t 

180L 


n  =  S.I4I6;    a  =60. 

u  =  2.2;  7t  =  S\. 


24.   — ;-  L  =  6;  a  =  10;     7t  =  S.I4I6. 


48  ALGEBRA. 


26.  -^  Tt  =  S,U16;     a  =  12;      r  =  10. 

27.^  a  =.8;  c=.100. 

q2     I     rt2  1V,2 

28.  ^  ^ — ^      «  =  4;  Z'  =  ic5;      ^  =  i5. 

29.  ^^^-^ a=z  10;  h  =  17]       c  =  21. 

K2 

30.  -TT  ^  =  ^;    ^^=  ^. 

h  ' 

31.  ;r(E  +  r)(K  -  r)  R-25\     r  =  20. 

32.  ih(L  +  l)  i  =  3.5\     1  =  85.25]     1=2^.75. 

33.  4:7ih{R  +  r)  2r  =  8;     2E  =  8. 

34.  7rh(R  +  r)(E~-r)  h  =  10;     E  =  5;    r  =  8.5. 

qK  i-r    ,-r*i-L;  \m=42;    h  =  9. 

(Eead  ^^  B  one/'  ^'  B  two  '^ ;  or  ^^ B  sub  one/'  '^  B  sub  two/') 

36.  jTrr^h  ;r  =  .^.i^i^;     r  =^  10;     li  =  8. 

37.  27rrh  n  =  S.U16;    r=z8;       h  =  7. 

{vZ. 


^*-    V-w  1  V=21. 


Q  -  110;  r=  5;  n  =  S.U16. 


39    -^ 

4r^7r 

40.   Trfab  +  -y  ]  a=^  18;     h  =  15. 


RA 

41 


a  —  A  \    r  =  15 


{^: 


=  i^6/;  A  =.125;  a  =  .875; 


^^-      a  -  A       ^^   ^  ^M  ^  =  100;    R'  =  110. 

(Read  "  R  prime  "  ;  or  ''  R  large  prime.") 
*  The  same  letter  can  be  used  to  denote  different  numbers  even  in 
the  same  formula  by  using  distinguishing  marks,  such  as  suffixes, 
as  in  35,  or  accents,  as  in  42. 


ABBREVIATION  OF  RULES.  49 

^^'  UA-W  c  =  S;     E  =  100;     R  =  21. 

Rr  +  R'r  +  R'R 
^^'  ^^'^ R  =  100;     r=120;     R'=1000. 

E 
45.  5 E=1.8;  x  =  2;  y=5;  R=  10;  r=100. 

1  /     a  b    \ 

46-  o -^ i;  a  =  .002;     b=.015;     n  =  60. 

2  W  —  a      n  —  b/  '  ' 

*'■   t fe  ~  ^J  T=8;     E  =  .75;     t=10. 

x(T  -  t)  (  ^^2^s;  T=16;  t  =  12; 

*"•  W(Q  -  T) 


I  a:  =  ;2^5;  T=  16;  i 
\Q  =  96;    W=1^5. 


D  f  c  =  .002;  t  =  20;  H=  60; 

l_|_tc  \h  =  S;  d  =  1.5;  D  =  13.5. 

(Q  -  t)(w  +  W)  |y=9.S;     ^  =  iO;      (2  =  ^6; 


Tf=5W;    r=i^(S. 


c,W(T  -  Q)       V    {  V  =  20;  w  =  S00;   W=10; 
"•  »  Ty(Q_t)        w    1  8^19;  t=10;  Q=1S.8;  T=20. 

W 

62.    ^(T  -  q)  -  s(r  -  t)  -  S(q  -  r) 

f  s  =  5;     S=1;W=500;      w  =  2Jfi;       t  =  0; 
\T=U;    q  =  6;     r  =  5.5. 

':i 
{ 


.,?„_,_.(T-t±i 


W=  1000;    w  =  25.6;    T=  100;    q=  20;    t  =  4; 
s  =  6. 
M 


J)  =  10;     d  =  1.6;     M  =  2. 


60 

56. 

67. 
58. 
59 

60. 


W(t,  -  2t,  +  t) 
d(W^  -  w) 
W- w 


W  +  yJ 


abc 
4K 
K 
s  —  a 


ALGEBRA. 

{P  =  12U)     W  =  2;     t^  =  U^S; 
\t^  =  11.5;        t  =  10. 

(w  =  S2;      W=  132;      W  =  1390; 
\  d  =  .995, 

V  =  500;      V  =  2000;     n  =  5. 

a=15;    1=28;    c=^l;    K=126. 
a  -\-  h  -\-  c 


s  = 


2 


a  =  19: 


h  =  20;    c  =  37;    K=  llJf. 


Formula  for  the  Area  of  a  Triangle. 

38.  An  extremely  important  formula  is  that  for  finding 
the  area  of  a  triangle  when  the  lengths  of  the  three  sides 
are  given  :  the  area  of  the  triangle  being  represented  by  K 
and  the  lengths  of  the  sides  by  a,  h^  and  c,  we  have  the 
formula 


Kz=^  Vsi^s  -  a)(s  —  h){s  —  c), 
where  s  represents 

a  -^-h  -\-  c 


EXERCISE  XXII. 


By  this  formula  find  the  area  of  the  triangles  whose 
sides  are  : 


1. 

a  =  12; 

l  =  H; 

c  =  25. 

2. 

a  =  25; 

^  =  51; 

c  =  74. 

8. 

a=  6   ; 

J  =  25; 

c=  29. 

4. 

a  =  16; 

Z>  =  25; 

c  =  39. 

6. 

a  =  17; 

h  =  25; 

c  =  28. 

ABBBEYIATION  OF  RULES, 


81 


Formula  for  Square  Root. 

39.   The  rule  for  square  root  can  be  conveniently  mem- 
orized by  means  of  a  formula: 

Ride,     I.   Separate    the    number    into 
5'  61'  69     2      periods  of  two  figures  each,  beginning  at 

__4 the  decimal  point.     Begin  with  the  left- 

161  hand   period,    subtract    its    square,    and 

bring  down  the  next  period. 
II.  Let  a  stand  for  10  times  the  part  of  the  root  already 
found.    Write  2a  for  a  new  divisor,  and  call  the  quotient  Z>; 
then  subtract  ^(2a  +  ^)  from  the  new  dividend. 


5' 61' 69 
4  00 


5(2«  +  ^')  =  1 


61 
29 


2Z=^a  +  h 


40  =  2a 

43  =  2a  +  ^> 


32 

III.  Bring  down  the  next  period  for  a  new  dividend. 
Let  a  stand  for  10  times  the  part  of  the  root  already  found, 
and  write  2a  for  a  new  divisor;  call  the  quotient  b,  and 
subtract  l{2a  +  b)  from  the  new  dividend. 


["40000  +  12900  =■]     j  I 
L52900  =  2302         J     (  ! 


5' 61' 69 
4  00  00 


61 

29  00 


h{%a  +  1))  = 


3269 


3269 


237  =  «  +  2> 


40 
43 
460  =  2a 

7  =  b 
467  =  2a  +  h 


0 


40.  This  process  can  be  shortened  by  leaving  off  the 
cipher  on  2a,  or  writing  it  lightly,  so  that  when  b  is  added 


52 


ALOBBRA. 


it  can  be  put  in  place  of  the  cipher;  then  the  completed 
work  would  read : 


Model  E. 


5' 61' 69 
4 

237 

1  61 
1  29 

43  =  2«  +  J 
3  =1 

3269 
3269 

467  =  2a  +  & 

1  =  h 

0 

Of  course  the  pupil  must  remember  that  at  each  new 
stage  of  the  process  a  new  meaning  is  given  to  the  letters 
a  and  h. 

EXERCISE  XXIII. 

Find  the  square  roots  of  the  following  numbers : 


1.  6889. 

2.  44944. 

3.  138384. 

4.  245025. 

5.  375769. 


6.  546121. 

7.  958441. 

8.  982081. 

9.  1948816. 
10.  1292769. 


11.  79.21 

12.  .049729 

13.  11.9716 

14.  .00710649 

15.  9545.29 


16.  811801. 

17.  3684.49 

18.  1.062961 

19.  418.6116 

20.  19796.49 


Approximate  these  square  roots  to  three  decimal  places : 


21.  2. 

22.  3. 


23.  5. 

24.  6. 


26.    10. 
26.    9.87 


27.  1.032 

28.  22.5 


29.  7.921 
80.  688.9 


EXERCISE  XXIV. 

Evaluate  the  following  formulce  : 


1.    V's(s  —  a)(s  —  b)(s  -  c)       a  =  29;  b  =  60;  c  =  85. 
2. 


^(3-a)(s-b)(s-^c)       ^^^^.  ^^^^.  ^^^^^ 


ABBREVIATION  OF  RULES.  63 


4^8(8  —  a)(s  —  b)(s  —  c) 
6.    |/r2  _  ^^2  ^  _  ^^^.  ^  ^  ^^5^ 


7.     |/li2+  ik2  ]c  =  ISO',  }l^7. 

2kr 


8.      ,  ^^^  r=^^;  k  =  9. 


1  ./"a^  (  1=25',  a  =  10;  h  —.981-, 

^^'  "21^  ;rr2k  (  n=  3,U16\   r  =  5',  h  =  8. 

11.  d|/^  t?  =  i.5;  ^  =r  ^^6?;  /=  .i5. 

^^^  ^(a  +  b)ab  ^  ^  ^      ^^^      ^  ^  ^^ 

Va2  +  b2 

^^-  '^'^  ;rld  I   Z  =  4;  ^  =  13.6. 

14.  '/2r2  -  r|/4r2  -  k^  r  =  100;  h  =  160. 

i/PvTTT^aT+t  (   a  =  273;    t  =  27;   D  =  . 00188; 

15.  f  ^(1.41)-^  \p^^^OJ^^,00. 

41.  Model  F. — In  the  following  formula  what  value  must 
be  given  to  n  when  the  other  letters  have  the  values 
T=  210,  iJ  =  85,  c  =  .0002  in  order  that  T'  may  have 
the  value  210.75  ? 

T'  =  {T-  n)+n{l  +  c{T-t)}. 


54  ALGEBRA, 

Let  the  required  value  of  n  be  represented  by  x]  then 

(210  -x)  +  x{l-\-  .0002(210  -  85)}  =  210.75 
210  -  a;  +  :?:{1.025}  =  210.75 
210 +  .025^  =  210.75 
.025a;  =  .75 
25a;  =  750 
a;=  30 

EXERCISE    XXV. 

__   P(l  +  k)  +  mi~k) 

^*  4h 

G  =  86.3895]  h  =  2-,  h  =  20,     Find  I. 

2.  G=i;rh[3(r,2  +  r,2)  +  h2] 

r,-=  JfO\  r,  =  SO;  h  =  SO;  G  is  foimd  to  he  131,880. 
What  value  was  used  for  n  ? 

3.  |(B  +  3T)  =  V 

When    a  =  3^0,    B  =  12,025,    and  V  =  1,615,000, 
what  must  he  the  value  ofT? 

*•  ^  y 

When  V  is  3000  and  v  =  15,  what  value  of  n  will  mahe 
D  ^  10? 

5.  ;r(ab  +  -^)  =  S 

What  value  must  he  given  to  a  in  order  that  S  may  he 
7854  when  h  is  50  f     {n  —  3.U16). 

The  Interest  Formula. 

42.  In  the  formula  j9r^  =  i,  p  stands  for  the  number  of 
dollars  in  the  principal,  t  for  the  number  of  years,  r  for  the 
rate  per  centum  expressed  as  a  decimal,  i  for  the  interest. 


ABBREVIATION  OF  RULES,  55 

Model  G. — Find  the  interest  on  $300  for  1  year  and  3 
months  at  5  per  cent. 

p  =  300;  r  =  .05;  t  =  1.25 
i  =  300(.05)(1.25) 
=  18.75  A?is.  118.75. 

43.   The  formula  for  the  amount  is  a  =p  -\-prt,  which 
can  also  be  written  j9(l  -\-  rt)  =  a. 

For  the  amount  at  compound  interest  the  formula  is 

a=p{l  +  ry. 

EXERCISE  XXVI. 

1.  What  is  the  interest  on  $532  for  3  years  1  month  and 
15  days  at  5  per  cent  ? 

2.  What  is  the  amount  of  $298  at  3|  per  cent  for  6 
months  ? 

3.  What  sum  of  money  will  return  $75  interest  at  2  per 
cent  in  2  years  6  months  ? 

4.  If  $1015  is  put  at  interest  at  7  per  cent,  how  long 
before  it  earns  $100  interest  ? 

5.  If  $784  earns  $15.68  interest  in  3  years,  what  is  the 
rate  exacted  ? 

6.  What  sum  of  money,  if  put  at  interest  at  3  per  cent 
for  5  years,  will  amount  to  $1000  ? 

7.  What  rate  of  interest  must  be  charged  on  $1500  for  5 
years  to  make  it  amount  to  $2047.50  ? 

8.  How  long  before  the  principal,  p,  will  double  itself 
at  5  per  cent  simple  interest  ? 

9.  How  long  will  it  take  $1500  to  amount  to  $1591.35 
at  3  per  cent  compound  interest  ? 

10.  Show  that  the  formula  for  r  in  compound  interest  is 


CHAPTER   III. 
TRANSFORMATIONS. 

44.  It  is  often  necessary  to  change  the  form  of  a  formula 
or  other  algebraic  expression,  or  to  perform  some  algebraic 
operation  upon  it.  In  order  to  be  able  to  do  such  things 
intelligently  we  must  investigate  the  laws  and  the  methods 
of  addition,  subtraction,  multiplication,  and  division,  in 
order  to  see  clearly  how  they  differ  from  the  similar  laws 
and  methods  of  ordinary  arithmetic. 

Negative  Quantities. 

45.  The  expression  a  —  b  is  the  algebraic  symbol  for  the 
result  of  subtracting  b  from  a.  But  if  b  is  greater  than  a 
the  subtraction  is  impossible.  The  question  is  in  that 
case  what  does  a  —  b  represent  ? 

46.  Take  a  particular  case.  Suppose  I  send  an  order  for 
10  quarts  of  berries.  If  the  grocer  has  a  quarts  in  stock  he 
will  have  after  filling  my  order  a  —  10  quarts. 

Suppose  his  original  stock  was  only  7  quarts.  He  would 
send  me  those,  but  his  condition  would  not  be  as  if  he 
never  had  had  any  berries,  nor  any  orders  for  berries;  for 
he  now  has  an  unfilled  order  for  3  quarts. 

To  put  these  circumstances  into  arithmetical  shape  : 

47.  It  is  required  to  subtract  10  from  7.  The  operation 
is   impossible.     But   7   can  be   taken  from  7,  leaving  0, 

56 


TRANSFORMATIONS,  57 

and  there  are  still  3  to  be  taken  from  any  number   that 

MAY    HEREAFTER  BE  ADDED    TO    THE    EXPRESSION;    jUSt   aS 

the  grocer  is  expected  to  fill  his  order  from  any  goods  that 
may  hereafter  come  in. 

0  7  -  10  =  -  3 

48.  The  expression  —  3  is  new  to  the  student.  It  means 
3  to  be  subtracted  from  any  number  that  may  hereafter  be 
added  to  the  expression. 

If  we  add  5  to  the  expression  in  0,  we  have 
®7-10  +  5=-3  +  5 

(D       -     3  H-  5  =  2 

If  we  add  3  to  the  expression  in  0,  we  have 
0-3+3=0 

49.  The  expression  —  3  differs  from  all  expressions 
known  to  arithmetic  in  this  particular,  namely,  that  you 
can  get  zero  by  adding  to  it. 

50.  When  two  quantities  added  together  give  zero,  one 
is  called  the  negative  of  the  other;  but  generally  by  a 
negative  quantity  we  mean  a  quantity  preceded  by  the 
minus  sign. 

That  is,  3  is  the  negative  of  —  3  just  as  much  as  —  3  is 
the  negative  of  3 ;  but  if  asked  which  of  the  two  was  the 
negative  we  should  say  —  3. 

51.  Terms  preceded  by  minus  signs  are  also  called  minus 
or  negative  terms ;  and  terms  not  preceded  by  minus  signs 
are  called  plus,  or  positive  terms. 

52.  Thus  it  may  be  said  that  any  positive  term  is  the 
NEGATIVE  of  the  Corresponding  minus  term ;  the  apparent 
contradiction  being  explained  when  we  remember  the  defi- 
nition. 


58  ALGEBRA. 

53.  Since  it  is  immaterial  in  what  order  two  numbers 
are  added,  we  may  as  well  add  —  3  to  3  as  3  to  —  3 ;  this 
would  lead  to  the  expression  3  +  (—  3)=::0,  where  the 
new  kind  of  quantity  is  added. 

64.  Finally,  we  may  translate  the  sign  —  by  the  words 
^'  the  negative  of." 

EXERCISE    XXVII. 

1.  li  a  -\-  h  =  0  what  is  the  negative  of  a^  ? 

2.  Find  the  value  oi  c  -\-  x  when  c  =  10;   x  =  a  —  I'y 

fl^  =  3 ;  J  =  5. 

3.  What  is  the  arithmetical  meaning  of  —  11  ? 

4.  What  is  the  negative  of  ~  11  ? 

5.  Translate  the  equation  —  (—  4)  =  4. 

6.  Simplify  -(-«);_(-  h). 

7.  Simplify  -  [-  a  -  h). 

8.  —  {x  —  a)',  simplest  form. 

9.  Find  the  value  of  x  -\-  y  -\-  a  when  x=  —  5;  y  =  2; 

a  ■=  c. 

10.  It  X  -\-  y  —  z  =  0,  what  is  the  negative  of  a;  ? 

Reciprocals. 

65.  When  the  product  of  two  numbers  is  1,  either  num- 
ber is  called  the  reciprocal  of  the  other. 

Thus  ii  ab  =  1,  a  is  the  reciprocal  of  J. 

The  reciprocal  of  7  is  |;  that  of  2^  is  f;  that  of  x  is 

-;  that  of  .25  is  4;  that  of  1.125  is  f ;  and  so  on. 

X 

66.  The  reciprocal  of  the  reciprocal  of  a  number  is  the 
number  itself. 

EXERCISE  XXVIII. 

1.  It  ah  =  1,  what  is  the  reciprocal  of  a  ? 

2.  If  5a;  =  1,  what  is  the  reciprocal  ot  x? 

3.  What  is  the  reciprocal  of  2.5  ? 


TRANSFORMATIONS.  69 

4.  \  =  a\  what  is  the  reciprocal  of  «  ? 

5.  What  number  added  to  its  reciprocal  gives  10.1  ? 

Rearrangements. 

57.  In  the  following  ^^  chain  of  additions  and  subtrac- 
tions '^  there  are  ten  terms,  some  positive  and  some  negative : 

5  +  17-3  +  2-30  +  15-3  +  5  +  2-1 

In  such  expressions 
I.  The  terms  may  be  rearranged  in  any  order,  without 
changing  the  value  of  the  entire  expression. 

II.  The  terms  may  be  grouped  in  any  way,  the  values 
of  the  groups  found,  and  all  the  results  added  together, 
giving  the  same  result  as  the  original  expression. 

That  is,  5  -  3  +  5  -  30  -  1  +  2  -  3  +  15  +  2  +17  =  9 
and  (5  -  30)  +  (17  -  3  +  2  +  15)  +  {-  3  +  2  -  1)  +  5 
or  -  25  +  31  +  (-  2)  +  5  =  9. 

EXERCISE   XXIX. 

Arrange  each  of  the  following  expressions  in  six  different- 
ways  without  changing  the  value  of  the  expression : 

1.  x+by  -\-  z.  4.  lOA  -  20^  +  21r. 

2.  X  -\-  y  —  z.  d.  z  -{-  2a  —  2b. 
S.  p  —  q  +  s. 

Make  a  group  of  ttvo  terms  in  each  of  the  following 
expressions  tuithout  altering  the  value  of  the  expression  ;  six 
answers  for  each  : 

6.  a  -\-  b  -{-  c  ~\-  d.  s.  ^x  -\-  2y  —  c  —  5d, 

7.  h  —  k  —  j  -\-  X,  9.  13^  —  31^  +  14r  —  4l5. 

10.   12:?:  +  23y  -  Slz  +  iw. 

58.  The  same  truths  hold  of  a  chain  of  multiplications 
and  divisions,  and  may  be  restated  as  follows,  understand- 
ing by  the  word  ^'  term  "  one  of  the  numbers  together  with 
the  X  or  -^  sign  preceding  it : 


60  ALGEBRA. 

I.  The  terms  in  a  chain  of  multiplication's  and  diyi- 
siOKS  may  be  rearranged  in  any  order,  so  long  as  each 
divisor  remains  a  divisor  and  each  multiplier  a  multiplier. 

II.  The  terms  in  a  chain  of  multiplications  and  divisions 
may  be  grouped  in  any  way  and  the  groups  multiplied,  so 
long  as  each  divisor  remains  a  divisor  and  each  multiplier  a 
multiplier. 

l~3xl5x3-^5x7-^14x3-h7x21-^2  X  6-r-9 
may  be  written 

1  ^  5  -f- 14-^7  X21x  7-^3xl5x2  x3-v-2x6-^9 

or      (1  -h  3  X  15  X  2)  X  (-^  5  X  7  -4-  14  X  3  -^  7)  X 

(21  -^  2  X  6  ~  9)  or  10  X  ^h  X  "^  which  equals  3;  under- 
standing that  -^  5  X  7  is  the  same  as  7  -^  5. 

EXERCISE   XXX. 

Arrange  the  folloiving  products  each  in  six  different  ways: 

1.  ahc,  5.  ^{x  —  y){a  —  b), 

2.  5ab.  6.  x{a  —  i)(x  —  z), 

3.  V/xy,  7.  {a  —  l)(l)  —  c){c  —  a). 

4.  ^x{a  —  h). 

The  purely  numerical  multiplier.,  or  the  coefficient  usually 
so  called,  is  generally  put  first ;  with  this  restriction  rear- 
range iji  six  luays  {including  the  order  give7i)  the  follo7ving : 
8.  bxy{a  —  c).     9.  21st{u  —  v).   lo.  301^(^  ~ j){j  —  ^)- 

Show  why  the  last  two  multipliers  in  10  are  not  alike, 

SUMMATION. 

69.  The  introduction  of  negative  quantities  makes  addi- 
tion and  subtraction  in  algebra  more  difficult  than  in 
arithmetic ;  and  the  necessity  of  having  terms  similar  before 
they  can  be  united  introduces  a  further  complication. 


TRANSFORMATIONS,  61 

60.  The  actual  operation  of  addition  in  arithmetic  con- 
sists in  taking  two  numbers  that  have  been  counted  up 
separately,  putting  them  together,  and  counting  them  up 
as  one  number.  For  example,  two  distances  are  measured 
from  the  beginning  of  the  first  to  the  end  of  the  first,  and 
from  the  beginning  of  the  second  to  the  end  of  the  second; 
when  they  are  added  the  beginning  of  the  second  is  put  on 
the  end  of  the  first  and  the  united  distances  are  measured 
from  the  beginning  of  the  first  to  the  end  of  the  second. 

61.  When  expressions  are  added  in  algebra  they  are 
written  one  after  another,  each  term  retaining  its  proper 
sign,  and  the  similar  terms  are  then  united  as  if  all  the 
expressions  were  one. 

Model  A. — Add  the  following  expressions: 

3a;  +  5  -  2^  +  1;  —  2 -\- 2x  —  10 -\- bx]  2;+ll  —  ^x+1. 
Operatio7i, — 

([)^x  +  b  —  2x  ^l  —  'Z  +  ^x  —  10-{-bx  +  x-\-ll 

-  3a;  +  1 
@  3:?;  -  2a;  +  2:z;  +  5a;  +  a;  -  3^  +  5  +  1  —  2  —  10 

+  11  +  1 
(3)  6a;  +  6 

The  form  @  came  by  rearranging  Q  according  to  I, 
§  58,  so  that  all  the  a;-terms  would  be  together;  then  (3) 
came  by  uniting  the  two  groups  of  terms  in  (2)  according 
to  §  21.  The  result  cannot  be  further  simplified  till  we 
know  the  value  of  the  letter  x, 

EXERCISE  XXXI. 

Add  the  following  expressions,  and  indicate  the  three  steps 
of  the  process  in  each  example : 

1.  3a;  +  2  +  4a;  +  7;  a;  +  3;  3a;  -  12;  1  -  10a;. 

2.  5a;  +  7a;  +  13a;  -  5a;;  3  -  17a;  +  2  —  3a;;  a;  —  5;  —  a;. 


62  ALGEBRA. 

3.   8  +  2:r  +  3  +  9ir  —  7  —  a;  --  4  —  3^;    2;  4    5  —  8a;; 

2a;  +  1;  —  x\  —  X. 
43^  5x  +  Ux  +  17  -  Sx  +  5  —  lOx;  4a;  +  2-3a;  +  7; 

34 -a;. 

5.  171a;  -  243  +  318.^'  -  411a;  -  111  +  150;  38a;  +  117; 

301  -  100a;;  102a.'  -  299. 

6.  3(a;  —  5) ;  5(a;  —  5) ;  8(5  —  x). 

7.  17(2a;  — 3);  11(4 -3a;);  (a;  -  1)3. 

8.  7a  +  341  +  2a  -  100  -  200«  -  200;   10{a  -  3); 

3(3  -  a), 

9.  3./  +  7;  (7  +  f/)3;  7(3 +  7/);   -(-3);  -3-7--y. 
10.    -(-35);   -(-8);   -(-3^-8);  d{s  +  8). 

62.  Only  similar  terms  can  be  united. 

EXERCISE   XXXII. 

Add  the  following  expressions: 

1.  3a; +  8;  3^  +  8;  2a; +  4?/;  a;  -y  -  15;  -5a;~3y  +  3. 

2.  2x  -\-  y  —  z;  x  —  3y  -\-  z;  4y  —  3a;  +  ^. 

3.  13a;  —  4:1/  -]-  z;  4y  —  10a;  —  bz;  4:Z  —  4:X  -{-  4:y  —  4. 

4.  -  131a;  -  100^  -  z;  -  {lOOz  -  100a;);  50a;  +  150y. 

5.  x  —  2y;  2y -\- x;  —x;  2x  —  y. 

6.  22)  —  3g;  2^-  q;  ^p  -2q;  p  +  q  +r  +  s. 

7.  dx  +  Uy  -(-   17);    20  -  10a;  -   lOy;   x-y-^l', 

-(-2). 

8.  10a;  +  r+2;;    a;  +  10r  +  c;    a;  +  r  +  10;2;;    x -{- r  +  z 

+  13. 

9.  dp  +  33r  +  3335;  2r  +  22s  +  222j9;  s  +  ll^j^  +  lllr; 

—  lOOr  —  IOO5  -  lOOp. 
10.  -1397^-128^;  h  -  g;    g  +  lOOh;    120k  -  h  -  2g; 
5^  +  5;^  +  5^  +  5. 

Subtraction. 

63.  In  subtraction  the    minuend  is  the  quantity  from 
which  the  subtrahend  is  taken  to  leave  the  remainder. 


TBANSFORMATlom,  63 

When  any  quantity  is  added  to  the  subtrahend,  the  efect 
on  the  remainder  is  precisely  the  same  as  if  the  same  quan- 
tity was  taken  from  the  minuend;  and  when  any  quantity 
is  taken  from  the  subtrahend,  the  effect  on  the  remainder 
is  precisely  the  same  as  if  the  quantity  was  added  to  the 
minuend.     For  instance,  the  following  subtractions: 

Minuend.  50  50-2  50  50  +  2 

Subtrahend.     30  +  2     30  30  -  2     30 


Kemainder.      20-2     20-2  20  +  2     20  +  2 

64.  It  is  possible,  therefore,  without  changing  the  value 
of  the  remainder,  to  take  any  term  out  of  the  subtrahend 
and  write  it  as  a  part  of  the  minuend;  only  we  must 
remember  to  change  its  sign,  from  +  to  — ,  or  from  —  to 
+.  And  if  we  can  so  transfer  any  term,  we  may  transfer 
them  all,  and  thus  convert  our  example  in  subtraction  into 
an  example  in  addition. 

Model  B. 

Minuend.     7:?:— 3^^+ 5   —ba—lOx^ 
Subtrahend.  ^  —  d>xy-{-2x-\-ba—  2x^ 
Remainder.    (To  be  obtained.) 
may  be  converted  into 
Minuend.      '7x—3xy-{-5  —  6a-'10x'^—3+Sxy—2x—5a+2x^ 

Subtrahend.  (All  these  terms  have  gone  into  the  minuend.) 
Remainder.  (New  minuend  simplified.)  6x-\-6xy-\-2  —  10a—8x^ 

65.  The  rule  for  subtraction  in  Algebra  is  therefore 
given  briefly  as  follows : 

Change  signs  in  the  subtrahend  and  unite  similar  terms. 

66.  To  save  time  in  calculation  the  signs  of  the  subtra- 
hend should  be  changed  IK  youk  mind,  and  the  terms 
united  without  writing  the  expressions  over  again. 


64:  ALGEBRA, 

EXERCISE  XXXin. 

Perform  the  following  subtractions  : 

1.  From  x^  —  7x^  +  16x  —  12  take  x^  ~  dx^  +  2x  —  48. 

2.  From  a  —  b  take  a  —  b  —  c, 

3.  From  x  —  y  take  y, 

4.  From  h  take  —  5. 

5.  From  x^  +  ^x^  +  2:^;  —  48  take  x^  -  4:X^  —  8x  +  8. 

6.  Take  x^  —  5x^  —  2^;  +  24  from  x^  +  2x^  +  Ax  +  3. 

7.  Subtract  24  +  Ux  -  29x^  +  6x^  from  2x^  +  dx^  -  13a: 

-12. 

8.  2  -  |/3  from  6  +  51^3. 

9.  Take  a  from  a  —  b. 

10.  From  9^  —  5^  +  4r  take  S^^  —  2^  +  2r. 

Parentheses. 

67.  To  save  using  words  like  subtract,  tahe^  etc.,  the 
operation  of  subtraction  is  often  indicated  by  putting  the 
subtrahend  in  a  parenthesis  with  the  sign  —  before  it. 
In  such  examples  the  parenthesis,  with  its  sigk,  serves 
only  to  mark  the  subtrahend  and  distinguish  it  from  the 
minuend. 

EXERCISE  XXXIV. 

1.  2a-2b  +  3c-d-  {6a  -  db  +  4c  -  7^). 

2.  x^  +  4^3  _  2^2  _|_  7^  _  1  _(^4  _[_  2x^  -  2x^  +  Qx-  1). 

3.  —  {a^  —  ax  -\-  x^)  +  3a^  —  2ax  +  x^. 

4.  4a;3-2a;^+:r+l-(32:3-a;2-2;-7)-0T3--4:i;2_[_2a;+  8). 
6.  lOa^'b  +  SaP  -  8a^^  -  b^  -  {ba%  -  Qab^  -  la^% 

6.  ^x^y  ~  Zxy'^  +  7^^  +  2;^  -  (8x^y  -  3xy^  +  9?/^  +  lla;^). 

7.  -JajS  —  \xy  -  1/  -(-  l:?:^  +  ^y  -  2/^). 

8.  \d'  -  ^a-1  -{-  1^2  +  t?  -  i). 

9.  f'^^^  -  i^  +  i  -  (i^  -  1  +  ^). 
10.  I^'?^^  —  -|^:f  —  (i  —  i^^  —  f«^). 


TRANSFORMATIONS,  65 

68.  Expressions  often  occur  in  Algebra  where  it  is 
desirable  to  take  out  parentheses  from  them  and  simplify 
by  uniting  terms.  To  do  so  it  is  necessary  to  remember 
first  that  a  parenthesis  with  —  before  it  indicates  subteac- 
Tiajq",  and  the  signs  of  all  the  terms  within  must  be 
changed  when  the  pakenthesis,  with  its  sign^,  is 
removed;  and,  secondly,  that  a  parenthesis  with  +  before 
it  indicates  addition,  and  no  change  of  sign  is  necessary 
when  THE  PARENTHESIS,  WITH  ITS  SIGN,  is  removed. 

69.  When  parentheses  occur  within  parentheses,  it  is 
best,  to  avoid  confusion,  to  begin  within  the  inner  one, 
that  is,  not  to  remove  a  parenthesis  which  has  another 
parenthesis  within  it. 

70.  The  sign  of  the  first  term  in  a  parenthesis  is  not 
the  sign  of  the  parenthesis;  thus  in  the  two  expressions 


Zz  —  x  —  y]   bz—{X'\-b\ 
the  sign  of  a;  is  +. 

EXERCISE  XXXV, 

Simplify  the  following  expressions : 

1.  a;  +  1  +  (5  -  a:)  -  (3  -  13^). 

2.  a;  +  2/  -  5  -  (:c  -  y  +  5). 

3.  2a  -  3Z>  +  (3a  -  U)  -  (2a  -  U), 

4.  a  —  l  -\-  G,—  {a  -\-  h  —  c). 

6.  14a  +  21h  -  13  -  (7a  -  llOh  -  17). 

6.  20;r  +  30y  -  16z  -  (I5y  +  16z  -  17x). 

7.  1108  +  135.^  -  780^  -  (39:?;  +  42y  -  109). 

8.  3^2  _|_  33^  _^  333  _  (333^2  _  33^  _|_  3) 

9.  2700^2  _  908^2  _  137^^ 

-  (1900a;  +  3000^;^  +  lOOy  +  1000^^)^ 
10.  x^  -\-  y^  -\-  2x  -\-  2y  —  {x^  —  y^  —  2x  -\-  2y). 
*  11.  2x  +  3y  -  5  -  9{5  -  y  +  x). 

*  See  §  13. 


66  ATMJSBnA, 

12.  a  -  bh  -\-2>c  +  {a  -  h)  -  2{6b  -  c  -  a). 

13.  X  —  y  —  6{'dy  —  172:). 

14.  2x  +  3.^2  +  1  +  2{x^  +  1)  -  5(1  +  x^). 

15.  4:{x  -  1)  -  5(2  -  3x). 

le,  a  -  b  -  3{2a  -  b)  +  7(a  +  b), 

17.  X  -]-  2y  +  {x  —  a)  —  2{a  —  3y). 

.18.  iz;  +  5  -  3(1  -  2:r). 

19.  x^  +  y'^  —  2{x  +  y). 

20.  «  +  5^  +  3c  —  ir  —  2(a;  —  a  —  5J  +  3c). 

Nests  of  Parentheses. 
71.  Model  C. 


a-[b-{c+{d-e-f)}] 
ci-[^~{c+{d-e+f)}] 
a^[b-  {c+  d-e+f}] 
a  —  [b  —    c  —   d  -{-  e  —f] 
a—    b -{-    c-\-    d  —  e -\- f 

EXERCISE   XXXVI. 

Simplify  the  following  expressions  : 

1.  X  -  {y  -  z)  +  x{y  -  z)  +  y  -  {z  -{-  x). 

2.  7a  -  [2b  +  \a-{b~\-  a)}]. 

z.  a  -  [3a  -  [bb  -  {4.c  -  Sa)}]. 

4.  [h-  {s  -  t)\  -^  [s  -  {t  -  h)}  -  [t  -  {Ic  -  s)] 

-  (h  +  s  +  t), 

5.  2x  —  (6i/  +  [4^  -  2:?:])  -  {^x  -  [?/  +  2;?]). 

6.  -  [  -  { -  (^  +  c  -  2«) } ]  +  [  -  I  -  (c  +  a  -  4^) }  ]. 

7.  -(-(-2a))-(-(-  (-3a;))). 

8.  4.x  -  [2x  +  2y  -  {x  -\-  y  -{-  z  -  {2x  +  2y -\- 2z  +  7c)]\ 

9.  -  3a  -  [4a;  +  {4c  -  {by  +  4a;  +  3a))]. 

10.  -  l^x  -  (12y  -  4a:)]  -  [6y  -  (4a;  -  7^)]. 

Where  the  answers  to  the  foUoiving  equations  are  not 
tvhole  nmnbers,  express  them  in  decimal  fractions  : 

11.  a;  +  3  -  (2a;  -  17)  =  4. 

12.  2a;  -  7  -  (a;  +  2)  =  2a;  -  (2a;  +  9). 


TllAN8FOltMATlONS.  6t 

13.  3^  +  2  ~  (2:?;  -  5)  +  (9  -  7:?:)  =  a;  +  (2  -  5^^;). 

14.  13  -  (5  -  2x)  -  (1  -  Ix)  =  25  -  (2  +  7x). 

15.  300  -  2x  -  (115  -  3^)  =  1000  -  (Sx  -  61). 

16.  100:r  -  (39  +  Sx)  -  10  =  12a;  +  345. 

17.  X  -  {Sx  -  2x  +  1)  r=  8  -  (3a;  +  7). 


18.  12a;  -  3(a;  -  2)  =  20  -  (5  +  9  -  7a;). 

19.  100  -  (2  -  15a;  -  23)  =  98  -  (20a;  -  24  -  5a;). 

20.  25a;  -  10(1  +  a;)  =  3        {x  -  5x  ~  7)  +  8a;. 

72.  To  put  terms  of  any  expression  into  brackets  similar 
precautions  are  necessary.  If  a  +  sign  is  used  before  the 
bracket,  no  change  is  necessary;  but  if  a  —  sign  is  used, 
that  takes  part  of  the  minuend  and  makes  it  a  subtra- 
hend ;  consequently  every  +  must  be  changed  to  — ,  every 
—  to  +.  Of  course  there  are  various  ways  of  bracketing 
the  terms  of  any  expression. 

Model  D. 

ax^  —  2/^0;^  +  ^y^  —  ^^^  ~  ^fy  +  ^  ^^J  be  written 
{ax'  -  2hxy)  +  {by'  -  2gx)  ^  {^  2fy  +  c) 
or         {ax'  +  hy'  +  c)  -  {2hxy  +  2^a;  +  2/^) 
or  {ax'  -  hxy  -  gx)  -  {lixy  -  ly'  -\- fy)  -  {yx  +fy  -  c) 
or  in  many  other  ways. 

EXERCISE  XXXVII. 

In  the  folloiinng  examples  get  tiuo  results,  first  with  a  + 
sign  hefore  each  bracket,  then  with  a  —  sign : 

1.  Bracket  terms  containing  x: 

lex  -f  cibx  +  CLcy  —  cibc, 

2.  Bracket  terms  containing  a  in  the  same  expression. 

3.  Bracket  terms  containing  «J  in  the  same  expression. 


68  ALGEBRA. 

4.  a^hc^  —  ah^c^  +  a/hh  —  aWc  —  a^hc^  +  aWc^  +  (^y^(?\ 
bracket  terms  containing  a^. 

5.  Bracket  last  two  terms :  W'c^  —  cPW'  —  aV. 

6.  Bracket  last  three  terms:  x^  —  y'^  -\-  %yz  —  z^, 

7.  Bracket  separately  terms  containing  a  and  terms 
containing  b:  ax  -\-  ly  —  ay  —  Ix  -\-  az  —  Iz, 

Bracket  like  powers  of  ^: 

8.  ax^  -\-  hx^  -{-  c  -\-  hx^  +  cx^  -{-  a  -\-  cx^  -\-  ax^  +  ^  + 
ax  -\-  hx  -{-  ex, 

9.  ax^  +  5a;3  -  aV  __  2^^3  _  3^2  _  j^4^ 

10.  ^2;^  +  ^^^^  —  ^^^  ~  ^^'^  —  ^^^« 

73.  The  letters  a^  Z>,  c,  J9,  and  q  in  the  following  expres- 
sions are  called  constants,  and  the  letters  x,  y,  and  z  are 
called  variables. 

EXERCISE  XXXVIII. 

Group  the  variable  terms  in  one  parenthesis,  obtaining 
t200  results  as  before^  one  with  a  +  bracket  and  the  other 
with  a  —  bracket, 

1.  a^  +  b^  +  x^  +  y^-  3a:^bx  -  2ab^x  +  ba%  -  '^x^y, 

2.  a^  +  c2  +  Z>2  _|_  x^  —  '^ax  -  2ab-2bc+2bc  +  2bx+2cx. 

3.  aV  +  2bx^-x^  +  2a^  +  4:P  -  2bx  -  a^  -  2ab  + 

4.  a{x  +  c)  -p{a  +  y)  -  c{x  -  y+p  -  q). 
6.  d{x  -  a)  +  5{a  -2x-  3y)  -  17{b  +  c-  2z). 

6.  2a{x  +  b-2c)-  2c{y  +q  -  2b), 

7.  bx{x  +  y  —  a)  +  lb{x  —  a  -\-  2p  —  6q)  —  Spq. 

8.  7a{2x  -p  +  2q)  -  3p{2y  -  a  +  2bc)  -  ba^, 

9.  2(4a  —  32:  +  2p)  -  b{2b  -  by-\-'dq)  -  7^1 
10.  I2x(a^  -  b^)  +  6a{a^  -  P)  -  2a{a  +  b-  2px). 


ax. 


TBANSFORMATIOm,  69 


MULTIPLICATION. 

74.  The  multiplier  and  the  multiplicand  are  called  the 
FACTORS  of  the  product ;  in  continued  multiplication  there 
are  more  than  two  factors. 

E.g.,  3  X  5  =  15;  15  X  7  =  105;  105  X  11  =  1155. 

3  and  5  are  the  factors  of  15;  15  and  7,  or  3,  5,  and  1, 
are  the  factors  of  105;  3,  5,  7,  and  11  are  the  factors  of 
1155. 

It  appears  from  the  principles  stated  in  §  58  that  the 
factors  of  any  product  may  be  written  in  any  order,  so  that, 
e.g.,  bah  =  bha, 

75.  When  more  than  one  of  the  factors  of  a  number  are 
alike,  it  is  sometimes  convenient  to  indicate  the  fact  as 
follows : 

2X3X3X3  =  2X3^  =  54;  ^aaaWbl  =  '^a%K 

76.  The  small  figure  denoting  the  number  of  equal  fac- 
tors is  called  an  index. 

77.  The  product  of  equal  factors  is  called  a  power. 

78.  One  of  the  equal  factors  of  a  power  is  called  a  root. 
For  instance,  27  is  the  third  power  of  3,  2  is  the  fifth 

root  of  32;  64  is  a  sixth  power,  2  being  the  root  and  6 
the  index;  36  is  a  second  power,  6  being  the  root  and  2  the 
index. 

79.  The  second  power  is  generally  called  the  square,  and 
the  second  root  the  square  root;  8  is  the  square  root  of  64. 

The  third  power  is  often  called  the  cube,  and  the  third 
root  the  cube  root;  5  is  the  cube  root  of  125. 

The  expressions  a^,  b^,  c^,  cP  .  ,  ,  p^  are  read  respectively 
a  square,  h  cube  (or  third),  c  fourth,  d  fifth  ,  ,  .  p  kth.* 

80.  Model  E.— Multiply  6x^^  by  dxY^^- 
bxxy.dxxxxxyy^yzz  is  what  we  get  for  the  factors  of  the 

*  Note  the  difference  iu  meaning  between  a',  a^,  a^  and  «i ,  aa,  o^s .  .  . 
(Read  a  two,  a  three,  a  four,  etc.) 


70  ALGEBRA. 

product.  These  can  be  rearranged  so  as  to  bring  the 
numerical  factors  together,  and  bring  all  like  letters 
together : 

5x3  xxxxxxx  yyyyy  zz 

which  can  be  simplified  thus : 

Ibx^y^z^, 

EXERCISE  XXXIX. 

1.  Multiply  ^x  by  4^.  4.  Multiply  a^  by  a?. 

2.  Multiply  'dxy  by  '7xy.  5.  Multiply  a^^  by  a. 

3.  Multiply  3abc  by  ac,  6.  Multiply  3a^b^  by  4a^Z>2. 

7.  Multiply  Sa^c  by  5a^c^. 

8.  Multiply  12a¥c^  by  16a^c\ 

9.  Multiply  ^a^c'  by  4:a^bc^.    12.  Multiply  2ahx  by  7aJ:r. 

10.  Multiply  a^  by  3«^.  13.  Multiply  xyz  by  15^^^;. 

11.  Multiply  d^x  by  aa:''^.  14.  Multiply  pqrs  by  7j9^ W. 

15.  Multiply  192;^y;2  by  lllx^y^z. 

16.  Multiply  ]  3ai3  by  7a'. 

17.  Multiply  301aV  by  2a^i2;l 

18.  Multiply  117 aWk^  by  ll^^^y. 

19.  Multiply  2naVz  by  112a^xh^K 

20.  Multiply  71^iV^  by  17^V. 

Multiply  together  the  folloiving  expressions : 

21.  x^y]  xy  ;  xy,  24.  S^r.'r^;  6a^xy;  6aPx. 

22.  i^^^^;;  a;^^^;  2;^^2;;  i?:^2;^.       25.  lOA^A;;  157^P;  2^^^^. 

23.  aWc;  ^^Z>6';  aZ>V.  26.  2a^^(?;  3a^bc;  4:a^bc, 

27.  lla^^^c;  12aZ^i2^;  6ab(^. 

28.  lOl^i^^V;  111^2^1  V;  13ai^Z^23^29^ 

29.  pqrs;  p^qrs^;  7pqhh\       30.  SOl^soi.  103/^103 ;  lOUi^^i^. 

81.  When  an  expression  is  separated  into  two  parts  by  a 
plus  or  a  minus  sign,  the  quantity  is  called  a  binomial. 

When  an  expression  is  separated  into  three  parts  by  plus 
or  minus  signs,  the  quantity  is  called  a  trinomial. 


TRANSFORMATIONS.  71 

When  an  expression  is  separated  into  two  or  more  parts 
by  plus  or  minus  signs,  the  quantity  is  called  a  polynomial. 

When  an  expression  is  not  separated  into  parts  by  plus 
or  minus  signs,  the  quantity  is  called  a  monomial. 

The  expressions  multiplied  in  the  preceding  examples  of 
this  chapter  are  all  monomials;  the  multiplication  of  bino- 
mials and  other  polynomials  depends  upon  the  multiplica- 
tion of  monomials. 

82.  Model  F.— Multiply  ba^  by  (3a;  -  2a), 

ba\^x  -  2a)  =  5d^ .  3x  -  6a^ .  2a  =  16a^x  -  lOaK 

EXERCISE  XL. 

Perfor7n  the  following  multiplications : 

1.  7x{x  -  1).  6.  2{x  -  5). 

2.  M\a  -  x).  7.  3A(4FA^-  -  2^2/2). 

3.  10h^k{2h  -  5F).  8.  10Wk'{lW^k^-21hW), 

4.  llp^qi^p^q  -  3prf).  9.  38hij{23hji  +  12ijh). 

5.  lxyH2xy'^  +  4a:'y).  lo.  10ax{100bx  +  lOOOcrr) 

Multiply  together  the  following  quantities : 

11.  3;  a;  -  5;  2.  13.  \0x^]  2y^',  ixy^  +  ^x^y, 

12.  3a(a  —  ^);  2x;  3ax.  14.  3a^;  a^  ~  P;  2ab, 

15.  lopq^;  9qr^;  i^f"^  —  -hV^- 

16.  7a^^;  ^La^p\  lbhq^\  i-^^P  ~  ii^M- 

83.  The  pupil  has  noticed  that  where  a  polynomial  is 
multiplied  by  any  number,  each  term  is  multiplied;  that 
is,  3{x  -  5)  =  3a;  -  15. 

84.  This  principle  may  be  illustrated  again  as  follows:* 
A  shopkeeper  sends  every  week  x  dollars  to  the  bank; 

his  messenger  uses  every  week  II  for  necessary  expenses; 
his  wife  draws  out  every  week  y  dollars  for  her  personal 
use;  his  son  remits  every  week  z  dollars,  which  is  added  to 

*  See  §  12. 


72  ALGEBRA. 

the  shopkeeper's  deposit.  The  increase  of  the  shopkeeper's 
account  each  week^  then,  amounts  to  x  —  1  —  y  -\-  z. 
In  5  weeks  the  increase  would  be  5(^  —  1  —  ?/  +  2;).  But 
in  5  weeks  the  shopkeeper  would  send  bx,  tlie  messenger 
would  use  up  5,  the  wife  would  draw  out  by,  and  the  son 
would  remit  bz.     Therefore 

b{x  —  1  —  y^z)  =  bx  —  b  —  by  -\-  bz, 

85.  This  is  the  third  of  the  important  and  fundamental 
principles  of  Algebra.     It  applies  also  to  minus  signs;  e.g., 

—  {x  —  z)  =^  —  X  -\-  z]   —  [x  —  y-\-z  —  a  —  h-^-c] 
=  — x-^y  —  z-{-a-{-b  —  c. 

It  may  be  stated  as  follows : 

The  product  of  a  polynomial  by  a  single  term  is  found 
by  multiplying  each  term  of  the  polynomial  successively. 

The  negative  of  a  polynomial  is  found  by  taking  the 
negative  of  each  term,  successively. 

86.  The  three  principles  stated  on  pp.  59,  60,  and  72 
are  called  the  thkee  fu:n^damektal  laws,  and  they 
are  known  as 

I.  The  Commutative  Law.     (§  57,  I;  §  58,  I.) 
II.  The  Associative  Law.     (§  57,  II;  §  58,  II.) 
III.  The  Distributive  Law.     (§  85.) 
Upon  these  laws  are  based  all  the  transformations  of 
Algebra. 

EXERCISE   XLI. 

Multiply  the  following  expressions : 

1.  bx{x  —  5  +  y);  ^V 

2.  ^x^y;   2x^y  +  12x^y^  —  Ibxy^  —  y*  —  4a;*. 

8.  —  {x^  -\-  y^  -\-  2^y) ;  ^^y- 

4.   -  2{W^h  -  daP  +  a^  -  b^);  4:a\ 
6.  WJc  -  217^3/^  +  42^*F;  b¥k\ 


TBAN8F0RMATI0NS.  73 

6.  lOpqr:,  bp^  +  3g^  +  7r^  +  '^p^q  +  ^q^  +  4^g^ 

7.  x^  +  4^^  +  5.T  —  24;  2a:i/^. 

8.  x^  +  r/:r^  +  Z>:?:^  —  ex  —  d\  abcdx^. 

9.  -  ?>li\%'lc  +  Ft  +  ^2y  +  jH  +  Fy  +  ph). 

10.    -   452^5^(75^  +  205^  -   155^  +   5^2  _!_  20i(3  _   15^4)^ 

Distributive  Factoring. 

87.  A  glance  will  show  when  a  monomial  is  a  factor  of 
every  term  of  an  expression,  and  such  an  expression  can 
therefore  readily  be  separated  into  factors.  Thus  in  the 
expression 

W  -  15aZ>  +  20^2  -  5« 

ha  is  a  factor  of  every  term,  and  the  whole  expression  is  the 
product  of 

ha(a  -  3^  +  4^2  _  i). 

EXERCISE:XLII. 

Find  the  factors  of  the  following  expressions,  or  show  that 
they  are  not  factorable : 

1.  ax  +  ay.  4.  3.r  —  15y.  7.  ^  +  x^. 

2.  ax  —  ay,  5.  a^  —  ah.  8.  6x  +  25a;^ 

3.  6x  +  lOi/.  6.  x^  -\-  x^y.  9.  13ic  +  91«^^. 
10.  i^^^y  —  ^y^.                      11.   7x  +  49ic2  +  343^:^ 

12.   3a;  +  6^;?/  +  2yz.         13.   13^^  _|_  55^2^  _^  117^:^1 

14.  343a;3y  +  9SxY  +  ^Sxy"^  +  Sy\ 

15.  15a;2  +  9a;y  +  25«/l    le.   382;3  +  67x^y  +  19xy\ 

17.  o;!^  +  lOx^y  +  45a;y  +  120a;y  +  210a;y. 

18.  45a;y  -  120a;y  +  210a;y  -  270^^^ 

19.  n^  —  71  +  n^  —  n^.      20.  ^5  —  2rh  -\-  rst^  -\-  rh^. 
21.  y^  —  2mxy  -\-  x  -{-  y.    22.  x^  —  2a;y  +  y^* 

23.  ic^  —  3a;^y  +  ^^y^' 

24.  ic*  —  4tX^y  +  6a;2^^  —  4a;y^  +  2/** 

25.  5a;y*  -  10x2^3  _[_  10^3^2  _  5^4^, 


74:  ALGEBRA. 

88.   The  process  of   ^^distributive  factoring ^^  is   often 
used  in  rearranging  terms  of  a  long  polynomial. 

Model  G.     x^—ax^  —  'bx^—cx^-\-ahx-\-acx-{-'bcx—al)c 
becomes,  when  we  bracket  like  powers  of  x, 

x^  —  (ax^-^^  l)x^  +  cx^)  +  {ahx  +  acx  -\-  hex)  —  abc, 
and  then,  factoring  each  parenthesis, 

x^  —  x'^{a  +  ^  +  c)  +  x{ah  -{-  ac  -\-  he)  —  ahc, 

EXERCISE   XLIII. 

Rearrange  hy  poiuers  of  x  the  first  four  expressions : 

1.  x^  —  d^'ji?  +  hx^  +  cji?  —  ahx  -\-  aex  +  he. 

2.  x^  \-  5a;^  —  4a;2  —  ^:ax  -\-  4^  -f  ax^  +  ^^-'^^  —  ^^^  —  ^^^« 

3.  cx'^  +  nx^  +  ^•'^^  +  ^"^^  +  ^^'  —  ^^^  +  ^^^ 

—  c?/LT^  +  cnx^  —  ^ma;^  +  ^^^* 

4.  2a;*  +  4:r^i/  +  ^x^y^  +  4:^^/^  +  ^*  -  ^xH  +  24:z;V 

-  32^^3  +  16^^ 

5.  From  «V  +  h^y''-  +  a^?/^  +  h'^x^  +  ^^^^  +  ^^^^ 

make  two  groups,  with  x^  and  y'^  for  coefficients. 

6.  From  the  same  expression  make    three   groups,  with 

(T,  V,  and  e^  for  coefficients. 

7.  In  the  sum  of  the  expressions  in  examples  1  and  3  find 

the  coefficient  of  x^, 

8.  In  the  expression  ax^  —  hx^  +  ^^^  ~  c^^  +  (^^  —  cib^ 

find  the  coefficient  of  a^  and  that  of  Z>^. 

9.  In  ax^  +  26«ir^  +  al?  +  27^x?/  +  llihy  +  2Azfx  +  2A^/^ 

+  hf  +  2^^^  +  ^t^'  +  2^^  +  2^^  +  2/y  +  2/^  +  ^ 
find  the  coefficient  of  x  and  of  y, 
10.  In  the  expression  a{bc  —  2ca)  +  ^(^^  —  2)  -{-c{ah—2,hc) 
find  the  coefficient  of  a^  and  of  a. 

89.  This  sort  of  factoring  is  of  especial  use  in  the  re- 
duction of  literal  equations. 


TRANSFORMATIONS.  75 


Multiplication  of  Polynomials. 

90.  In  multiplying  polynomials,  the  multiplicand,  consid- 
ered as  one  quantity,  is  multiplied  by  each  term  of  the 
multiplier;  then  each  of  these  partial  multiplications  is  car- 
ried out. 

Model  H.  (5«  +  3Z>)  (2«  -  7^>)  =  5a(2a  -  7Z>)  +  U{2a  -  U) 
=  10a2_35aZ>+6aZ'-21Z>2=10a2-29a^-21Jl 

Model  I.  {2a  —  hx){'^a  —  ^x)  ='^a(^a  —  4:x)  —  bx{2>a—4:x) 
=  6«2  —  ^ax  -  {15ax  -  20x^)  =  Ga^  -  4.ax  —  16ax  +  20^^ 
=  Qa^  -Wax  +  20:6-1 

Model  J.     {a-\-d~c){a—h-\-c)  =  a{a—I)-i-c)-{-b{a—b-{-c) 
—  c{a  —  h  -\-  c)  —a^  —  ah  -\-  ac-\-  ab  —d^-{-bc—{ac—bc+  c^) 
=rt^  —  ah  -{- ac  -\-  ah  —  W-\-hc  —  ac-\-hc—c^ 
^a^-h^-^'lhc  -  c\ 


EXERCISE  XLIV. 

Multiply : 

1.  2x—y;  2y+x.  11.  a^+ah-{-h^;  a—h, 

2    p^+3pq+2q^;  Ip  —  bq.  12.  p^—q^]  p^-\-q^, 

3.  p^-pq-^q^\p^^n^(f'  13.  li^-lc'^  ¥+¥k^+]c\ 

4.  Ji^~lik+21c^\¥+]i]c-2¥.  14.   x-\-y-\-z]  x-^y—z, 
6.  x^^%xy-\-y'^\x^~\-2xy—y'^.  15.  a^-^-^th"^-,  a-\-4th. 

6.  x^~4:]  x^-\-b.  16.  x^—2x^-\-S\  x-\-2, 

7.  x^—^x-^^]  x^—Qx-\-b.  17.  a  —2h-\-c]  a-{-2h—c, 

8.  0:^+52:— 3;  x^—bx—S,  18.  a^—7a-\-5;  a^~2a-{-3. 

9.  a;+15;  x—S.  19.  3x—6y;  dx^—iy^. 
10.  x^  —  dx+2;  x^-3x^+2.  20.   20+:x;;  a;2-10. 

Simplify  these  expressions  : 

21.    {dx'^  +  2x  +  l){3x  -  5).  22.  (x^  +  5x+  25){x  -5). 

23.   {x^  -7x-  U){x  +  11).  24.   {dx^+4x+5){Sx-10). 


76  ALGEBRA. 

25.  (6a;2  -^x-{-  2)(2.'?;  +  1).  29.  {U^  +  lU-\-  49)(2if~7). 

26.  {^x^-%xy-y'){'ix~y),  30.  {^lf-QU'H'd){^t-\-'i). 

27.  {x^+?^xy+4:y'^){^x-4.y),  31.  (16-28i?+49jt?2)(4+7i?). 

28.  (:z;^+2:^''+4:^;+8)(^-2).  32.  (38^2_i9^2)(2^2_|_y2)^ 

33.  {^x^  +  2x  -  6){2x^  +  5x  -  1). 

34.  {^x^  +  4ic2^  —  3xy'^){5x^  —  4:r^  +  Sy^). 

35.  {x^  +  ^x^  +  9x  +  27)  (^2  —  2x  —  3). 

36.  C'^^  -  2x^  +  4:X  -  8){x^  +  X  -  2). 

37.  {x^  +  32;  +  9)(^  -  3)  -  {x^  -  2x  +  4){x  +  2). 

38.  (m^  +  n^  -  1)2  -  (m*  -  ^^  _^  1)1 

39.  {x^  +  2x  -  Sy  -  {x^  ^  2x  +  3)2. 

40.  (17a2  ^  iQab  +  15Py  -  {Ua^  +  16aZ>  -  17Z>2)2. 

The  Law  of  Signs. 

91.  It  is  convenient  to  notice  a  law  in  regard  to  the  sign 
of  the  separate  terms  of  the  product. 

Where  any  term  of  the  multipher  is  plus,  the  partial 
product  obtained  by  it  has  for  each  of  its  terms  the  same  sign 
as  the  corresponding  term  in  the  multiplicand;  where  any 
term  of  the  multiplier  is  minus,  the  partial  product  ob- 
tained by  it  has  for  each  term  the  sign  opposite  to  that  of 
the  corresponding  term  in  the  multiplicand. 

To  express  the  four  possible  cases  in  a  table: 

Multiplicand.  +     +     —     --' 

Multiplier.  +     —     +     — 

Product.  +     —     —     + 

92.  From  this  table  the  following  law  is  easily  derived  : 
In  multiplication  like  signs  give  plus  and  unlike  minus. 

93.  The  same  rule  holds  for  division,  as  will  hereafter 
be  shown. 


TBAN8F0RMA  TI0N8, 


77 


Cross-Multiplication. 

94.  Simple  products  like  {x  —  b){x-\-  2)  have  important 
features. 

Model  K. 


I    4 


i 


-10 


'^^ 


-j^ 


^ 


'% 


X  —  ^ 

a;  +  2 


H^l 


^^1  J>^ 


ic2  -3a;  —10 

The  products  of  terms  directly  under  each  other  are 
called  STRAIGHT  PRODUCTS ;  of  terms  diagonally  opposite 
each  other  cross  products.  The  straight  products  are 
not  similar,  and  so  cannot  be  united  ;  the  cross  products 
are  similar  and  can  be  united. 

In  this  example  the  straight  products  are  x'^  and  -—  10; 
the  cross  products  are  —  ^x  and  +  ^^^  and  their  sum  —  Zx\ 
the  entire  product  is  x^  —  Zx  —  \^, 


EXERCISE  XLV. 


In  the  foUoiving  examples  name  the  straight  products^  the 
cross  products,  the  sum  of  the  cross  products,  and  the  en- 
tire product : 


1. 

{x  +  2){x-7). 

16. 

L. 

(11-^70(2  + A). 

2. 

(«  +  5)(«  +  9). 

17. 

(4^  -  \){U  -3). 

3. 

(a  +  3)(a-10). 

18. 

{%x  +  3)(2:r  +  3). 

4. 

{b+7){b-6). 

19. 

(3y  -  7)(3^  +  7). 

5. 

(/i  +  3)(A  +  8). 

20. 

(65  +  l)(55  +  10). 

6. 

{h-ll){h+2). 

21. 

(^  +  2.y)(a;  -  ^y). 

7. 

[h  -  l){h  ~  4). 

22. 

(a-\-  hl)){a-\-U), 

8. 

(^  +  10)(a;  +  10). 

23. 

{3a  +  2Z>)(5a  +  2J). 

9. 

(y-3)(y  +  3). 

24. 

(c  +  i^)(3c  +  2^). 

10. 

{s  +  l)(s  +  99). 

25. 

(3/(;-4(/)(5/^  +  ^). 

11. 

(2^  +  l)(7^-l). 

26. 

(IO2;  -  2/^)(32;  +  570. 

12. 

(5a  +  l)(9«  +  l). 

27. 

(4^>  -  3>10(3Z^  -  2^). 

13. 

(3a  +  2)  (5a  -  2). 

28. 

(3:r  +  5^)(3.^  +  5iy). 

14. 

(c  +  l)(3c+3). 

29. 

(3^  -  ll.)(3^  +  11.), 

15. 

(*-7)(5yfc  +  l). 

30. 

(65  +  50(105  +  70- 

/^  mc^  of  the  folloiving  examples  ascertain  tvhat  vahie 
X  must  have  in  order  that  loth  expressions  may  be  equal ; 
where  the  answers  are  not  whole  numbers,  express  them  as 
common  fractions  : 

31.  {x  —  6)(x~-^);    {x—2){x  —  7). 

32.  {x  -  2){x  -  8) ;  {x-  3){x  -  4). 
83.   {x'-7){x+l);   {x  +  3){x-6). 

34.  {x+8){x-5);  (:«;-10)(a;-  8). 

35.  {x-2){x  +  4);  (^^  +  ll)(.r  -  4). 

36.  (22; -5) (^-2);   {2x+5){x-2). 

37.  l2x-3){x  +  12);  (22:  +  l)(.T-ll). 

38.  {2x  +  7)  {2x  -  5) ;  {ix  -  6){x  -  5). 

39.  (3cc- 5)(7:?;  +  2);   {3x  -  5)(lx+ 3). 

40.  (17^  -  29) (^  -  3);  (17:^;  -  29)(:?;  +  2). 

6.   Products  like  those  here  formed  by  cross-multiplica- 
tion are  called  quadratic  expressions. 


TRANSFORMA  TIONS, 


79 


Factoring  by  Cross-Multiplication. 

96.  Model  L. — In  the  product  x^  —  ^x  -\-  14  the  straight 
products  are  x^  and  14  and  the  sum  of  the 
cross  products  is  —  ^x.  The  terms  that  give 
the  straight  products  must  have  like  signs, 
and  therefore  the  cross  products  must  have 
like  signs;  both  cross  products,  then,  are 
minus.  To  get  the  first  straight  product  we 
have  XX  x\  to  get  the  second  2x7;  then 
the  signs  must  be  chosen  so  that  both  cross 
products  are  minus;  that  gives  us  {x  —  2) (re  —  7). 

EXERCISE    XLVI. 

Name  the  straight  ^products  and  the  sum  of  the  cross  prod- 
ucts i?i  the  follotving  expressions,  and  find  their  factors : 


1.  cc^— 5.T+6. 

2.  x^—2x-\-l, 

3.  :«;2_3^_}_2. 

4.  X^  —  4:X'{-3, 

5.  X^-\-4:X-{-4:. 

6.  x^-\-5x-^4t. 

7.  x^+7x-\-12. 

8.  x^—7x-^6. 

9.  x^+7x+10. 

10.  x'^—8x-{-l^. 


11.  x^Sx+7. 

12.  x^-\-Sx-]-15. 

13.  .^'2-20:^;+19. 

14.  x^-20x+Q4:, 

15.  x^+20x-{-'d6. 

16.  ir^+202;+51. 

17.  x'^  —  20x+'76. 

18.  cc2_^20x+84. 

19.  x^—14:X-\-4c9. 

20.  a;2— 14a;+13. 


21.  x^—14:X+33. 

22.  2;2— 14^4-24. 

23.  X^-]-14:X+4:5. 

24.  2;2-f  14^-f48. 

25.  x^-8x+16. 

26.  ^2-121a;+120. 

27.  2;2-121a;+238. 

28.  x2-121^+1210. 

29.  2;2-121^+3550. 

30.  a;2— 10002;+999. 


^ 


97.  Model  M. — In  the  product  x^  —6x  —  24:  the  straight 
products  are  x^  and  —  24  ;  —  5x  is  the 
sum  of  the  cross  products. 

The  terms  giving  x^  for  a  product  had 
like  signs,  and  the  terms  giving  —  24  for 
a    product    had    unlike    signs  ;    then  the 
two  cross  products  had  unlike  signs,  and 
the  minus  cross  product  was  the  greater.     To  give  x^  for  a 


W'^    __- 


80  ALGEBRA, 

straight  product  we  should  have  to  multiply  x  and  x]  to 
give  24  we  could  multiply  1  and  24,  2  and  12,  3  and  8, 
4  and  6 ;  and  in  each  case  the  larger  number  would  have  to 
be  minus. 

x  +  l  i»  +  2  a;  +  3 

X  —  '^^  x—1^  x—  d> 


-\-x  —  '^4:X  +  2^  —  l%x  -\-^x  —  ^x 

Trying  each  in  succession,  the  first  pair  of  factors  to  give 
the  right  cross  products  is  {x  -\-  'd){x  —8). 

EXERCISE    XLVII. 

Nayne  the  straight  products  and  the  sum  of  the  cross 
products  m  the  following  expressions,  and  wherever  pos- 
sible find  their  factors : 


1. 

X^  —  X~2, 

11. 

^2-26:^-407. 

21. 

Sa:^-x-10. 

2. 

x^-x-Q. 

12. 

^2_4^_437^ 

22. 

2x^+'dx-9. 

3. 

X^-dX-4:, 

13. 

x^+4:X-221. 

23. 

bx^-9x—2. 

4. 

x^+x-Q, 

14. 

.'^2_10.^'-231. 

24. 

6x^+x-15. 

6. 

x^+2x-l5. 

15. 

^2_  14^-6840. 

25. 

10:^2-9:^—9. 

6. 

x^+3x—2S. 

16. 

2:2-282:-1100. 

26. 

15.t2-72;-2. 

7. 

X'^-4:X-21. 

17. 

^^2-1-91:?;- 900. 

27. 

Ux^+llx-15. 

8. 

x^'-4:X—32, 

18. 

^2_j_9Q^_1944^ 

28. 

2l2;2+:?;-10. 

9. 

x^+3x-70. 

19. 

:?:2+ir-10302. 

29. 

10:?;2_29^_21. 

10. 

x^+4.x-96. 

20. 

x'^-54.x-3(j27. 

30. 

6x'+17x-U. 

98.  In  the  equation  2:r  —  7  =  0,  2;  must  have  the  value 
■I;  and  if  we  substitute  this  value  for  x  the  expression 
2x  —  7  becomes  zero. 

Model  N.     6a;2  —  5a;  —  6  factors  into  (3a;  +  2)  (2a;  —  3). 
To  make  3a;  -f  2  =  0  we  must  have  a;  =  —  f ; 
To  make  2a;  —  3  =  0  we  must  have  x  =  ^, 


TRANSFORMATIONS.  81 

EXERCISE    XLVin. 

What  value  7nust  we  suhstitute  for  x  in  each  factor  of 
the  folloioing  expressions  in  order  that  the  factors  may 
severally  hecome  equal  to  zero  ? 

1.  x^  -  3x  +  2.  6.  Ux^  +  5:?:  -  1. 

2.  x^  -  6x+  4.  7.  2lx^  -  Ylx  +  2. 

3.  ^x^  —  X-  2.  8.  3.^2  -  14a;  +  16. 

4.  "^x^  -  3ic  +  1.  9.  Vlx^  -7:^+1. 

5.  x^  —  X  —  2.  10.  15^:2  —  ll:r  —  14. 

99.  This  sort  of  factoring  is  of  especial  use  in  the 
solution  of  quadratic  equations;  it  is  also  used  with  other 
devices  in  the  reduction  of  fractional  expressions. 

Arrangement  in  Multiplication. 

100.  In  multiplication  it  is  often,  if  not  always,  conven- 
ient to  arrange  the  work  somewhat  as  in  Arithmetic,  with 
the  partial  products  on  separate  lines,  as  follows* 

Model  0.  Model  P. 

x^  —  xy  -\-  3^2  X  —  h 

X  —  "ly  X  -\-  "l 


x^  —  x^y  +  3^y^  ^^  —  5x 

-  2xhj  +  2xy^  +  6y^  ^       +  2x  -  10 


x^  —  3x^y  +  5xy^  —  6y'^  x^  —  3x  —  10 

101.  The  advantage  of  this  arrangement  is  that  similar 
terms  in  the  different  partial  products  may  be  more  readily 
seen.  A  further  help  in  this  direction,  for  long  examples, 
is  the  arrangement  of  multiplier  and  multiplicand  ^^ac- 
cording to  powers  '^ ;  that  is,  taking  for  the  first  term  of 
each  that  with  the  highest  power  of  some  one  letter,  for 
the  next  term  the  next  power  below,  and  so  on;  e.g., — 

4:5xY  +  210a;y  -  lOxif  +  210^4^^  +  x^"^  -  120^^^^ 
+  Uxy  +  y'^  -  1202;y  -  10^%  -  252^y 


82  ALGEBRA, 

could  be  arranged  according  to  powers  of  x  thus : 

*  +  2\0xHf  —  UOxy  +  210:^;2?/«-  10a;/+  ^^^^ 
or  according  to  powers  of  y  thus : 

yio  _  iQ^y9  _|_  45^,2^8  _  i20xhf  +  2102:4^^  _  252a;y 

+  2102;y  -  120a;y  +  210xY-10xhj+x^^ 

EXERCISE  XLIX. 

Rearrange  the  folloiving  expressions  hefore  multiplying : 

1.  {a^x^  +  x^-\-  a^){a^  +  x^  -  aV). 

2.  {x^  +  y^  +  ^y){y^  +  ^^)- 

3.  (a'  +  ^a'h''  +  ^6  +  Wa?){a'  -  Z^«  +  Sa^^^  _  3^2^4)^ 

4.  (V  -  Z^5+  5^^^-  ba'l)  +  lOa^b'^-\Oa%^){d'-\-h^-'Uh), 
6.   (6a;V  +  2;^4-a^  —4:X^a  —  4:Xa^)(^a^x  —  ?>ax^~a^  +  a:^). 

6.  (W  +  /^4+54)(A2_^2), 

7.  (5^2  _  2.Ty  +  32;2)(2:^?/  +  S^;^  -  5/). 

8.  (3/iF  +  ¥  -'^Wh  -  ¥){¥+k^-QJi'k^-^¥lc-A:h¥). 

9.   (21/^2_|_35y^3_|_35y^4_|_7^6,^_^y_|_^7_|_21^/^5_j_  7^^6) 
(y_|.^3_|_3^2^_j_3^^2), 

10.  (i^^^— 6^^y  + 132;^— 12^^^+  4?/^)  (o;^  —  ^'^  -]-xy^  —  y^). 

11.  (^'+l^  +  4)(|-i^  +  ^'). 

12.  (3a2  +  ^-2«)(5^2__  1  _  1^), 

13.  (a;2  +  a:y  +  iy^){y^  -^y  +  x% 

14.  (fr2-i:.+  5)(^:,.  +  ^^2„^). 

15.  {j\a^  +  2^2^  +  -%U^  +  |a&2)(A^2  _^  1^2  ^  2aZ>). 

DIVISION. 

102.  In  division  we  are  given  one  facror  of  a  number  to 
find  the  other.  The  given  factor  is  the  divisor,  the  re- 
quired factor  is  the  quotient,  and  their  product  is  the  divi- 
dend. With  regard  to  the  signs  of  the  separate  terms  of 
the  quotient,  they  must  be  such  that  when  multiplied  by 


TRAN8F0RMA  TI0N8.  8  3 

the  terms  of  the  divisor  they  produce  the  signs  given  in  the 
dividend. 

To  express  the  four  possible  cases  in  a  table  : 

Divisor.  +     +     —     — 

Dividend.       +     —     +     — 
Quotient.        _{____[- 

103.    Hence  the  rule: 

In  division  like  signs  give  plus  and  unlike  signs  minus. 

EXERCISE  L. 

Divide : 

1.  12x^  ly  2x.  5.  ^Sx^a  ly  l^ax. 

2.  Mx^  ly  4:x\  6.  400a^%^  by  IQab. 

3.  9x'  hy  ^x\  7.  na^hVi^h  by  ISaWhh. 

4.  4^5  by  5x\  8.  d25a^'z^  by  Ida^hK 

9.  {a  +  by  by  {a+bf, 

10.    '72a'{Il^  -  P)10  Jy  8^3(^2  _  p)7^ 

11.  abc-^  {-  c).  -  d4.'da^{a  -  by 

12.  l^ah  ~  3ah.  ^^'      -  49(a  -  by   ' 

-  39axY  -  76{¥  -  Uy 
^^-     Uax^y^'                          ^^'      19[h^-U)    ' 

—  348a;y  17.  333^}'^  -r-  (—  9ik), 

^^-    -  V2xY  '  18.   -  469a{b  -  c) -^  27a. 

19.  1002^(5  -  c)3  -r-  {  -  6ib  -  cy\. 

20.  (-400)  --  (-  3.2). 

21.  12a^  +  Qab  +  30a  by  6a. 

22.  2a  +  Uah  +  6a^hJc  -  26a^  by  2a. 

23.  21xYz^  —  4:9xYz^  —  dx^z^  by  3xyz. 

24.  3ij  +  Qi^h  —  9H  +  12ijk  by  3i. 

25.  -  3a¥h  +  6aPF  +  9ah¥  by  -  3ahh, 

26.  l^m^n^  —  42m^^^  +  28mn'^  —  3bmhi^  by  7mn. 

27.  ISJc^i^  +  1)  +  9h\]c^  +  1)  ~  12/^(7^  +  h)  by  -  3k. 

28.  2^(5  +  ty  -  6t{s  +  ty  +  sk{s  +  ty  by  {s  +  ty. 


84  ALOEBRA, 

29.  r(r2  +  s^)  +  5(r2  +  s^)  -  t{r^  +  s^)  by  -  {r^  +  s^). 

Long  Division. 

104.  Model  CI. — Divide  x^  —  y^\)^  x  —  y. 

The  correspondence  of  multiplication  and  division  may 
be  seen  as  follows.  The  arrangement  given  for  division  is 
recommended  as  the  most  compact. 


Divisor.              ^  —  y 
Quotient.            x^  +  ^y  +  y^ 

Multiplicand.          ^  —  y 
Multiplier.               x'^-\-xy-\-y'^ 

Dividend.           x^  —  y^ 
1st  subtrahend,  x^  —  x^y 
x^y  -  y^ 
2d  subtrahend,  x'^y  —  xy'^ 

1st  partial  product,  x^  —  x^y 
2d  partial  product,  x'^y  —  xy'^ 
3d  partial  product,  xy'^  —  y^ 
Product.                 x^  —  y^ 

xy^  -  y^ 
3d  subtrahend,  xy'^  —  y^ 

0 

EXERCISE  LI. 

Divide : 

1.  x^  —  y^  by  X  -\-  y, 

2.  x^  +  ^^y^  +  y^  iy  x^  —  xy  -\-  y^. 

3.  x^  -\-  x^y  4"  ^^y^  +  cc'y^  -\-  xy^  +  y^  by  x^  +  y^- 

4.  Sc^  -  22c^x  +43^2^:2  -  3Scx^  +  llx'^  by  2c^-3cx+4:X^ 

5.  x^  -  3x^  +  9a;  -  27  by  x^  +  9. 

6.  a^  -]-  b^  -\-  c^  —  3abc  by  a  -\-  b  +  c. 

7.  s^  -25^  +  1  by  s^-2s  +  1. 

8.  x^  —  xV  —  x^  ~{-  a^  by  x^  —  xa  —  x  -{-  a, 

^      ^6  —  ^3^3  _  ^3  _|_  ^3  J^y  1^3  _^  ^2  _  ^2^  _|_  ^  _  ^5  —  ^^ 

10.  h^""  +  JiY  +  ^'^  by  Iv"  +  hg  +  g^. 

11.  (K  +  ^)  -  (i^  +  !)• 

12.  (f«^  -  \ab'  -n^)-^  {la  +  b). 

13.  (1^4  +  |«&^  -  m  -^  («  +  1^). 


TRANSFORMA  TI0N8,  8  5 

14.    (K  +  l«^^^  -  ^^^'^  -  l«'^')  -^  (-2^  +  «^)- 

16.  (K + i«'^' + m  -  (i^'  +  \^^  +  i^'). 

TF^ere  ^^ere  is  a  remainder  in  the  folloiving  divisions, 
write  the  remainder  as  the  numerator  and  the  divisor  as  the 
denominator  of  a  fraction  which  is  one  term  of  the  complete 
quotient : 

16.  {x^  +  y^)  -^{x-  y). 

17.  (x^^  +  x^y^  +  y^^)  -^  {x^  +  2/5). 

18.  (a^  +  a%  +  \ah^^^^)^{^---a\ 

19.  1  {x^  -  yy  +  xy{7?  +  y^  +  dxy)  \  ^  {x^ -\.  xy -^^  y^)- 

20.  (3a*  +  27a^3  -  10^*)  .^  (^  _  2^>). 

EXERCISE  LII. 

Perform  the  operations  indicated  in  the  following  expres- 
sions : 

1.  {a'  +  h^){a  -  h){a^  +  ah  +  V'). 

2.  {a:'  +  ^ah  +  h^){a'  -  2ab  +  P). 

3.  {a^  +  a%  +  aV"  +  W){a  -h). 

5.  (a3+3a2Z>+3aZ>2_^Z>3)(a+J)-(6-a)(a34-a2^4-^52_|_j3)^ 

6.  [a^  +  3a2(Z>  +  6?)  +  3a(^>  +  cf  +{b  +  cf]  ^{a-\-h  +  cf. 
x^(y  +  z)  -{-  y^{z  -\-  x)  -\-  z^{x  -j-  y)  —  x^  —  y^  ~  z^  —  2xyz 

2;  -  (^  -  ^)  • 

h\k  +  s)  +  h^s  +  h)  +  s\h  +  lc)-\-  Uhs 

ks  -\-  sh  -{-  hk 
(g2  -{-st^  t^){s^  -st  +  t^){s^  -  t^)  -  {3sH^  -  3g¥) 

{h^  ~U  +  9){¥  +  6h  +  26){¥  -2h-  15) 
{h^  -  8^2  +  24.h  -  45) 

(^*  +  a;^^/  +  xY  +  xf^  +  ^4)(:r2  +  i?;y4-y2)(^_^y 

{x^-y'^Y  +  ^xY  ci^  -  3a^Z>  +  3a^>^  -  Z>^  +  g^ 

a;2  —  ic^  -4-  ^2     •         •  ^  _  ^  _|_  ^ 


7. 

8. 

9. 
10. 
11. 


86  ALGEBRA. 


EXERCISE  LIII. 

Substitute : 

1.  ^  =  c?;  +  2  in  the  expression  dx  -\-  4:y  —  25. 

2.  ^  =.  2y  —  3  in  the  expression  3a;  -f-  3;2;  —  21. 

3.  ;^  =  3  —  ^  in  the  expression  ^  +  10  —  5^. 

4.  y  =  2x  -{-  3  in  the  expression  o;^  +  3xy  +  ^^« 

5.  X  =  y  —  1  in  the  expression  3^  +  2xy  +  2a;^. 

6.  ;2  =  1  —  2/  in  the  expression  y^  -i-  z^ -^  3xy  -^  y  -{-z, 

7.  y=2  — 3^;  in  the  expression  ir(y— 3)— ^(2;— 3)+^'^^^+^^. 

8.  ^  =  2y  +  7  in  the  expression  x^  +  xy  -\-  2y^. 

9.  y  =  —  x^  in  the  expression  xy  -{- 1, 

10.  y  =  a:^  +  a;  +  1  in  the  expression  y  +  xy  —  x\ 


11.  a  = 

:X^ 

-12a; +  16;  Z>  = 

=  a;3-12a;-16: 

;  c  = 

a;2-16 

m  — . 
c 

12.  ^  = 

:X^ 

—  2a;  +  l;  k  = 

a;3  _  3^2  ^.  3^  . 

-  1 

m  -^^ — 

-3x 
k 

+  2) 

13.  h  = 

a' 

+  a;3;  c  ^  a^  + 

,     2-     K«' 
(T^o;  +  a;^  m  -7-5- 

c{a^  • 

-f  ax 
—  ax 

14.    S  = 

X^ 

-  4a;3a  +  6a;V  - 

-  Axa^  +  a';    t 

^^ 

■  -  2x^a 

+  2xa^  - 

■a' 

.    s{x'^  +  2aa;  + 

a^) 

16.  X  =  a  -\-  b  -{-  c;  y  =  {ab  -\-  be  -\-  ca)  in  \-l* 

16.  s  =  a^\  t  =  be  ]n  -^ '—— . 

s  -{-  t 

17.  x-\-Z  for  2;  in  the  equation  4:Z^  +  9^^  =  24;^. 

18.  a;  —  5  for  z  in  the  equation  .2^  =  lOz  —  ?/^. 


TRANSFORMATIONS,  87 

19.  ;2  +  5  for  y  in  the  equation  IQx^  —  2by^  +  200xy  =  0. 

20.  h  —  1  for  X  in  the  equation  3^^  —  ^xy  =  1y  —  %hy, 

21.  a—1)  for  y  in  the  equation  %a^  —  y'^  =  y{a-]-b)  +  2a^. 

//i  each  of  tlie  following  examples,  ivhat  value  must  x 
have  in  order  that  the  given  expressions  may  he  equal  to 
zero  ? 

x^  -  11:^:  +  14  x^-  14^2  ^  17^  _  (5 


22. 
24. 
26. 
26. 
27. 
28. 
29. 
30. 
31. 


+  2a;-  7  *  ic^  +  Sa;^  -  5^;  +  2 

x^  -  n       * 

3a;5  -  16:^;4  _  35^3  ^  2a;  -  14 
^x^  +  5^3  +  2  '• 

6a;^  -  15a;^  +  4:^;^  +  4^;^  +  :^;  -  2 

62;^  -  3^=^  -  2a;2  +  1 
6:z;5-  33:^;^  -  10^+55 

3:^4-5 
45a;5  -  261a;^  -  260:^;^  -  33^  +  220 

15^*  +T3:?  -  11 
blx^  -  ^bx^  +  63^2  -  171a;  +  110 

19a;^  +  21a;  -  22 
6a;^  —  4a;^  -  16a;^  +  lOa;^  -  9a;  +  15 
•  2a;4  +  2a;3  -  2a;2  -  3  ° 

81a;5  +  306a;^  +  33a;^  -  6a;  -  22 
27a;^  +  3a;3-2 


CHAPTER   IV. 
IDENTITIES   AND   THEOREMS. 

105.  In  regard  to  the  following  equations: 

I.   3^  +  5  =  22;  +  7  II.   3(.^;  +  5)  =  3^  +  15 

the  pupil  will  notice  that  while  the  first  equation  is  true 
only  for  the  particular  value  x  =  2,  the  second  is  true  for 
any  value  that  may  be  chosen.  Again,  that  the  expression 
3^;  +  5  cannot,  by  any  means  we  know  of,  be  transformed 
into  2:?;  +  7;  while  3{x -\-  6)  is  in  a  very  simple  way  trans- 
formed into  3x  -\-  15. 

106.  These  two  kinds  of  equations  have  distinct  names. 
An  equation  which  is  true  only  on  condition  that  the  let- 
ters in  it  have  particular  values  is  called  an  equation  of 
condition;  while  an  equation  which  is  true  for  any  values 
whatever  of  the  letters  in  it  is  called  an  identical  equation. 

Equations  of  condition  are  the  equations  ordinarily  met 
with  in  solving  problems. 

Identical  equations  (or  equations  of  identity,  or  simply 
identities)  may  be  recognized  by  the  fact  that  one  member 
can  be  transformed  to  the  identical  form  of  the  other 
member.  The  two  members  are  said  to  be  identically 
equal,  and  sometimes  the  sign  =  is  used  instead  of  =. 

The  Proof  of  Theorems. 

107.  A  Theorem  is  a  general  statement  requiring  proof. 

Theorems  in  algebra  are  often  proved  by  stating  them  as 


IDENTITIES  AND   THEOREMS,  89 

equations,  and  showing  that  one  member  can  be  trans- 
formed so  as  to  be  exactly  like  the  other;  that  is,  showing 
the  equation  to  be  identical. 

108.   Model  A. — Prove  the  following  theorem: 
The  product  of  the  sum  and  difference  of  two  numbers 
is  equal  to  the  difference  of  their  squares.* 

Proof, — Let  a  and  d  represent  any  two  numbers,  then 
the  theorem  is  expressed  by  the  following  identity : 

{a-^l)){a-  h)^a^-l)'^. 

In  transforming  the  first  member  we  get  for  straight 
products  a^  and  —  ^  ^ ;  the  cross  products  are  ah  and  —  ah, 
and  their  sum  is  zero ;  so  that  the  entire  product  is  a^  —  h'^, 

EXERCISE    LIV. 

Prove  the  folloiving  tlieorems  : 

1.  The  square  of  the  sum  of  two  numbers  is  equal  to 
the  square  of  the  first  number,  plus  twice  the  product  of 
the  two,  plus  the  square  of  the  second. 

(The  identity  is  {a  +  hf^a""  +  lah  +  V.) 

2.  The  square  of  the  difference  of  two  numbers  is  equal 
to  the  square  of  the  first  number,  minus  twice  the  product 
of  the  two,  plus  the  square  of  the  second. 

3.  The  square  of  any  polynomial  is  equal  to  the  sum  of 
the  squares  of  the  separate  terms,  added  to  twice  their 
products,  taken  two  at  a  time. 

(The  straight  products  are  the  squares;  show  that  the 
cross  products  are  double.) 

4.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  one  more  than  double  the  less  number. 

(Let  a  be  the  less  number,  ^  -j-  1  the  greater. ) 

5.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  equal  to  their  sum. 

*  This  tlieorem  will  be  hereafter  referred  to  as  Theorem  A. 


90  ALGEBRA, 

6.  The  sum  of  the  squares  of  two  consecutive  numbers 
is  one  more  than  twice  their  product. 

7.  The  difference  of  the  cubes  of  two  consecutive  num- 
bers is  one  more  than  three  times  their  product. 

8.  The  sum  of  the  cubes  of  two  numbers,  divided  by 
the  sum  of  the  numbers,  is  equal  to  the  sum  of  the  squares 
of  the  two  numbers  minus  the  product  of  the  numbers. 

9.  The  difference  of  the  cubes  of  two  numbers,  divided 
by  the  difference  of  the  numbers,  is  equal  to  the  sum  of 
the  squares  of  the  two  numbers  plus  the  product  of  the 
numbers. 

10.  The  product  of  three  consecutive  numbers  is  equal 
to  the  difference  between  the  middle  number  and  its 
cube. 

(Let  a  —  1,  a,  and  a  -\-lhQ  the  numbers.) 

11.  The  product  of  two  consecutive  numbers  is  equal  to 
the  smaller  number  plus  its  square. 

12.  Prove  Theorem  A,  letting  a  be  the  smaller  of  the 
two  numbers,  and  h  the  difference  between  them. 

13.  Prove  the  same  theorem,  letting  a  be  the  larger  of 
the  two  numbers,  and  h  the  difference  between  them. 

14.  The  difference  of  the  squares  of  two  consecutive 
even  numbers  is  twice  their  sum. 

15.  The  difference  of  two  numbers  formed  by  the  same 
two  digits  in  opposite  order  is  always  divisible  by  9. 

16.  The  difference  of  two  numbers  formed  by  the  same 
three  digits  in  opposite  order  is  always  divisible  by  99. 

109.   Identities  may  be  translated  into  theorems. 

Model  B.— In  the  identity  {a-^h)'^  =  a^+b^+3al){a  +  l)), 
a  and  h  stand  for  any  two  numbers,  because  an  identity  is 
true  for  all  numbers;  a  ^  h  is  the  sum,  a^  and  b^  are  the 
cubes,  ab  is  the  pkoduct  of  any  two  numbers.  So  the 
theorem  may  be  stated : 

The  cube  of  the  sum  of  any  two  numbers  is  equal  to  the 


IDENTITIES  AND   THEOREMS,  91 

sum  of  their  cubes  plus  three  times  their  product  multi- 
plied by  their  sum. 

Similarly  for  the  identity 

(a  -  hf  =  a^-¥-  ^al){a  -  h). 

Many  algebraic  theorems  are  so  complicated  that  they 
can  be  conveniently  stated  only  as  identities.  Such  are 
the  following. 

EXERCISE    LV. 

Show  that  these  identities  are  true : 

1.  {a^  +al)  +  ¥)  (^2  -ah  +  b')  =  a' +  a'}?  +  IK 

2.  «5  +  Z>^  =  (a  +  l){a'  -  a%  +  a^W  -  ah^  +  ¥). 
^  a^-.h^=\a-  h){a'  +  a^h  +  a^l)'  +  aW  +  b% 
4.  a'-b'=  (V  ^  ^2)(^  _^  ^)(^  _  j^Y 

6.  {a  +  h-^  cf  =  a^  +  ^a^{h  +  c)  +  ^a  {b  +  c)^  +  {b+c)'. 

6.  la  +  by  E=  a^+  4.a^b.+  QaW  +  4.ah^  +  b\ 

7.  {a  +  b-\-  c)  {be  +  m  +  ab)  =  d\b  +  c)  +  h\c  +  a) 
+  c\a  +  b)  -\-  dabc. 

8.  ^«  +  y^  =  {x^  +  y^){x^  -  xY  +  y^)* 

9.  a;«  -  /  =  {:^  +  ^) (cc  -  y){:x^-  xy  +  y^)  (a;2_|_  ^.^  _|_  ^2)^ 

10.     0>^  +    ^>15  =  (^3  _|_    ^3)  (^12  _    ^9^3   _L    ^6^6  _    ^3^9   _[_   J12) 

APPLICATIONS   OF    THEOREM   A. 

110.  Theorem  A  is  by  far  the  most  important  of  those 
here  given.  Its  application  leads  to  one  or  two  convenient 
processes  in  Arithmetic.  In  Algebra  its  applications,  to  be 
hereafter  shown,  are  of  even  greater  importance. 

Finding  Squares  by  Theorem  A. 

111.  By  Theorem  A  the  square  of  any  number  can  be 
readily  calculated  when  the  square  of  any  number  near  it 
is  known. 


92  ALOEBBA, 

Model  C— To  find  the  square  of  23  : 

23  +  20  =  43 
23  -  20  =  3 


129  :=  23^  -  202 
232  =z  529 


Model  D.— To  find  59^  : 

60  +  59  =  119 
60  -  59  =  1 

119  =  602  -  592 
.-.   3481  =  592 

EXERCISE  LVI. 

Find  in  this  way  the  squares  of  the  following  numbers : 

1.  45.         3.  78.  6.  29.  7.  249.  9.  988. 

2.  37.         4.  117.         6.  198.         8.  1011.  10.  793. 

A  somewhat  different  method  is  as  follows  : 
Model  E.— To  find  the  square  of  23  : 

23  +  3  =  26 
23  -  3  =  20 


520  =  232  -  32 
529  =  232 


Model  F.— To  find  592 : 

59  +  9  =  68 
59  -  9  :=3  50 

3400  =  592  -  92 
.-.   3481  =  592 

In  applying  this  theorem  the  pupil  must  try  to  find  some 
number  which,  added  to  or  subtracted  from  the  given  num- 
ber, will  make  an  easy  multiplier. 


IDENTITIES  AND  THEOREMS.  93 

EXERCISE  LVII, 

By  the  second  mstJiod find  the  square  of: 


1.  111. 

3.  98. 

5.  85. 

7.  1998. 

9.  288, 

2.  57. 

4.  52. 

6.  49. 

8.  311. 

10.  63. 

EXERCISE    LVIII. 


Find  by  Theorem  A  the  difference  of  the  squares  of  each 
of  the  folloiving  pairs  of  numbers : 


1. 

3  and  73. 

6. 

121  and  120. 

2. 

9  and  109. 

7. 

8133  and  8131. 

3. 

575  and  425. 

8. 

2731  and  269. 

4. 

339  and  319. 

9. 

101  and  99. 

5. 

1723  and  277. 

10. 

10001  and  1. 

The  pupil  may  with  profit  invent  arithmetical  illustra- 
tions for  the  other  theorems. 


Factoring  by  Theorem  A. 

112.  Wherever,  in  the  product  of  two  binomial  factors, 
the  cross  products  disappear,  the  factors  will  be  seen  to  be, 
respectively,  the  sum  and  the  difference  of  the  same  two 
quantities,  or  multiples  of  them ;  so  that  such  a  product  is 
also  an  illustration  of  Theorem  A. 

Model  G. — For  example,  I'^x^  —  8  is  the  product  of 
^x  —  ^  and  62;  +  4,  which  may  also  be  written  3^  —  2 
and  2(3:c  +  ^)?  ^^^  i^  ^^^^  ^^  original  expression  might 
have  been  written  2(9:^;^  —  4)  and  the  parenthesis  factored 
by  Theorem  A:  since  it  is  the  difference  of  the  squares  of 
^x  and  2,  it  is  the  product  of  their  sum  (3^;  +  ^)  ^^^  their 
difference  (3:^;  ~"  ^)' 


94 


ALOEBBA, 


EXERCISE   LIX. 


Factor  the  foUoimig  expressions : 


1.  4.^2  -  9. 

2.  8^:2  -  162. 

3.  18^2  -  32. 

4.  27x^  -  147. 

5.  1728a;2  -  12. 

6.  49a;2  -  16y\ 

7.  4aW  _  i21a;2. 

8.  dQxY  -  25;^^ 

9.  1007^^  -  36P. 
10.  1210«3Z,3  _  iQab^ 


11.  243a;^V  —  l^x^z. 

12.  fl^^^^  —  96'^. 

13.  aV  -  49. 

14.  1  —  X^'^.     «- 

15.  4  —  a^ 

16.  9  —  16xY- 

17.  1  -  lOOA^F. 

18.  75a;^o  __  48^8^ 

19.  9a^'c^  -  9x^\ 

20.  1  -  100a'h'c\ 


21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


z  4-  y)'^  —  ^^. 
z  —  yY  —  x^. 
X  +  y)2  -  4. 

X  +   1)2   -   6«l 

x-\-  2^)2  -  ^l 
2a;  +  3«)2  -  9/. 
:^  +  6a)2  -  4^4^ 

2a;  -  5.^)2  -  ^c\ 
9a;  —  d^y  —  a^. 


31.  x^  —  {y  -\-  zy, 

32.  x^  —  {y  —  zy. 

33.  4  —  (a;^+  ?/)l 

34.  ^2  —  (x  +  1)2. 

35.  ^2  -  {x  +  2yy, 

36.  9/  -  (2a;  +  3a)2. 

37.  4/  -  (^  +  6^)2. 

38.  49^2  _   (^2  _^  7)2^ 

39.  9^2  -  {2a;  -  5^)1 

40.  a^  —  (9a;  —  a2)2^ 


What  relation  exists  between  the  expressions  iiumhered  21 
and  31,  22  and  32,  23  and  33,  e^c.  .^  PTAa^  relation  exists 
between  their  factors? 


41.  {a  +  J)'^  -  (^  +  ^)'. 

42.  {a  —  by  —  {x  —  yy, 

43.  {x  —  ^)2  —  {a  —  by, 

44.  (a  -  hy  -  (Z>  +  ^)'. 

45.  (2a;  -  yy  -  {x^  +  2^)2. 

46.  {a  —  3a^)2  —  (4y  —  2^)2. 


47.  {a  +  2by  -  (4a;  +  5^)1 

48  («^  +  by  -  {3a  +  2>2)2. 

49.  {x^  —  x^y  —  {x  —  1)2. 

60.  (V  -  a3Z>)2  -  (a2^2  _  ^^3)2^ 

51.  a;2  4- 2a;y  +  ^^  —  («^  +  ^)^- 

52.  a2  —  2a J  -]-P—{x  —  yy. 


IDENTITIES  AND   T1IE0BEM8.  95 

53.  9^2  -  Qah  +  Z>2  _  {^x  -  3^)2. 

64.  a^  -  2ah  +  h^  -  {x^  +  2z)^ 

65.  «^  +  2a^  +  r^  -  {iy  -  zf. 
56.  €?  -  {V  -  Uc  +  ^2)^ 

67.  4a^  —  {y^  —  2yz  -\-  z^), 

58.  c^  -  (25^2  -  dOab  +  9Z>2). 

59.  {x  -  yf  -  («2  +  Qah  +  W). 

60.  x'^  +  ioax  +  25^2  _  [iQz^  +  8;2  +  1). 

61.  9.^2  -  4^2  +  12«Z>  -  W. 

62.  1  -  «2  -  2a^'  -  Z>1 

63.  16:?;^  —  x^  —  14:xy  —  49«/^. 

64.  a^  —  2ah  -\-V^  —  x^. 

65.  9c^  -  4^2  +  4.ai  -  P. 

66.  x^  -\-  2xy  +  ^^  —  «^  +  2(^^  —  P, 

67.  tt^  -  2ab  -^V  -  c^  -  2cd  -  d'\ 

68.  x^  —  ^ax^-\-  ia^  —  1)^  -\-2hy  —  y'^, 

69.  4.^2  _  i^i^  ^  9^2  _  9.^,4  _!_  30^2^2  _  25^,4^ 

70.  x^  +  2x-^l-  a'  -  2d^h^  -  ¥, 

113.  In  the  ten  examples  last  preceding  the  groups  of 
terms  that  must  be  bracketed  together  to  make  a  perfect 
square  are  easily  found;  but  even  where  carelessly  arranged 
they  may  be  picked  out  by  selecting  the  cross  products 
first,  then  the  straight  products  ;  and  by  remembering  that 
in  a  perfect  square  the  straight  products  are  +,  and  conse- 
quently the  negative  straight  products  must  appear  in  the 
negative  brackets,  i.e.  the  subtrahend,  the  second  of  the 
two  groups  into  which  the  expression  is  to  be  divided. 

Model  H. — In  the  expression  x^  —  x^  —  ^—  2aV  +  a'^-\-6x 
2a^x^  and  6x  are  the  cross  products,  and  the  two  groups  are 
x^  —  2aV  +  a^  and  —  2;^  +  6^  —  9,  of  which  the  second 
must  go  in  the  —  bracket. 
x^  -  2aV  +  a^  -  {x^  -  6x  +  9)  =  {x^  -  a^f  -  (x  -  3)2. 

The  factors  of  this  expression  are 

{x^  -a^  +  x  -  3)0^2  -  a^  -  x  +  ^). 


96  ALGEBRA, 

EXERCISE  LX. 

Factor  the  following : 

1.  x^  -{-  y^  -^  2xy  —  4:X^y^, 

2.  ia'^b'  —  Sab  —  a^  —  \m. 

3.  a^  -  9aV  -  iab  +  U^ 

4.  x'^  —  a^  -{-  1  —  2x  —  4:ab  —  ^b^. 

5.  ^a  -  Mx-\-l  -  x^  -  166^2  _^  9^2^ 

6.  x^  —  a^  ^  y'^  —  ¥  —  2xy  -\-  lab. 

7.  ^x?  -  12ax  -  c^  -  ¥  -  2cJc  +  9al 

8.  a^  +  ^bx  -  Wx^  -  \<dab  -  1  +  2W. 

9.  a''  -  2hx^  +  %a^x^  -  9  +  30^;^  +  \^xK 
10.  a^  -  a'  +  4^2  _  J2  _^  2aZ>  +  4. 

/S^o?7i6  (?/*  ^/^e  folloioing  examples  have  more  than  tivo  fac- 
tors : 

11.  {a^  -  'dVf  -  4^2^^  16.  (62;2*-  35^2)2  _  ^2^2^ 

12.  (2^2  -  3^2)2  _  ^2^2.  ^^  (7^2  _  11^)2  _  36^6^ 

13.  {x^  +  12)'^  -  492^2,  18.  {^x^  -  10^/2)2  _  ^2^2, 

14.  {x^  -  5x)2  -  16^>l  19.  (3.^2  +  xtj)'^  -  lOOy. 

15.  {li^  -  ^hf  -  36.  20.  (67^2  _  177^^)2  _  9^2^ 

Completing  the  Square. 

114.  If  we  know  that  an  expression  is  the  square  of  a 
binomial,  we  need  know  only  two  of  its  terms ;  the  missing 
term  can  be  constructed  from  those  two  by  the  use  of 
Theorems  1  and  2.* 

Model  I. — Thus  if  we  know  that  9^^  ^nd  49  are  the  first 
and  last  terms  of  a  perfect  square,  we  know  that  the  terms 
of  the  binomial  are  ^x  and  7;  and  the  binomial  must  be 
then  either  3:?;  +  7  or  3:?;  —  7 ;  in  the  first  case  the  middle 
term  is  +422;  (twice  their  product),  and  in  the  second  case 
the  middle  term  is  —  42^;. 

*  See  p.  89. 


IDENTITIES  AND   THEOREMS. 


97 


The  complete  square,  then,  of  which  ^x^  +  49  are  the 
two  straight  products,  would  be  either 

^x^  +  42a:  +  49  =  {^x  +  7)^ 

or         2x^  -  42x  +  49  =  (3^  -  7)2. 

Model  J. — Again,  if  we  know  that  x^—  24:X  are  the  first 
two  terms  of  a  perfect  square,  the  last  can  be  found  by 
remembering,  according  to  Theorem  2,  that  while  x^  is  the 
square  of  the  first  term  of  the  binomial,  24:?;  is  twice  the 
product  of  both  terms;  so  the  second  term  of  the  binomial 
is  —  12;  and  the  last  term  of  the  square,  then,  would  be 
144.  The  complete  square,  then,  of  which  x^  —  24^  are 
the  first  two  terms,  is  x^  —  24a;  +  144  =  (^  —  12)^. 

115.  The  same  facts  may  be  more  clearly  seen  by  con- 
sidering the  diagrams  for  cross-multiplication.  In  the 
first  case  the  double  cross  product  is  missing;  in  the  sec- 
ond case,  since  the  middle  term  is  known  to  be  a  double 
cross  product,  the  single  cross  products  are  each  half  of  it. 


EXERCISE     LXI. 

Complete   the  square  implied  in  each   of  the  following 
expressions : 

5.  x^  —  18r?;^ 

6.  49^2  +  625. 

7.      9X^  —   24:X. 


8.  64^2_|_4ooy. 

9.  4:X^  +  729. 
10.   16:^;^  —  200x^y, 


1.  x^  +  112a;. 

2.  x^  +  2dQ. 

3.  x^  —  144a;. 

4.  a;^  +  16. 

116.    We  are    sometimes  enabled,   by  means  of    these 
facts,  to  transform  an  expression  so  that  it  can  be  treated 


98  ALGEBRA. 

as  the  difference  of  two  squares,  and  factored  by  Theorem 
A.  Thus  Q^x^  +  625  becomes  a  perfect  square  by  addhig 
400a:^;  and  if  we  also  subtract  400^;^  we  obtain  the  original 
expression  unchanged  in  value,  but  transformed  so  as  to  be 
readily  factored  by  Theorem  A. 

Model  K.     Ux^  +  625  =  Ux^  +  4:0Qx^  +  625  -  4002^2 
^(8:?;^  +  25)2-  {20xy 
=  (8:z;2-20:z;+25)(82;''^+20iC+25). 

EXERCISE   LXII. 

In  the  same  way  factor : 

1.  x^  +  4.  6.  4x^  +  1296;2l  8.  324^^4  +  625. 

2.  4:^4  +  81.         6.  642;*  +  81.  9.   1024^*  +  81^. 

3.  x^  +  324.         7.  2500:^4  +  1.  10.  ^x^  +  6561. 

4.  4.x^  +  625f/4. 

117.  Some  trinomials  can  also  be  treated  in  this  way. 
This  is  the  case  when  the  number  to  be  added  to  the 
middle  term  to  make  it  a  double  cross  product  is  itself  a 
perfect  square. 

Model  L. — Factor  x^  +  x^y"^  +  yK 

x^  +  x^  -\-y^  =  x^-\-  2xY  +  y^  -  ^y 
=  {x^  +  y'^f  -  {xyf 
=  {x^  +  y^  +  xy){x^  +  y'^-  xy). 

EXERCISE   LXIII. 

In  the  same  way  factor : 
1/^4  _|_  9^2  _^  81.  3.  x^  +  xhf  +  y^.     6.  1  +  ^x^  +  ^\x\ 
2    x^  +  x^  +  1.        4.  16a;*  +  4x^  +  1.    6.  I62:*  +  36x^  +  81. 

7.  x^+xY+y^  (3  factors).      13.  x^  —  "Ixhf  +  /. 

8.  256c«;8  4-  16:^;*  +  1.  14.  x^  -  2x^  +  1. 

9.  ^5  _^  9^3^2  _|_  81^J4^  j5^    ^4  _  27^2^,2  ^  J4^ 

10.  48^;*  +  12^2  _[_  3^  jg^  16^4  „  24a2  +  1. 

11.  x'  -  6x^  +  1.  17.  81^*  +  8a;2  +  16. 

12.  x^  -  WxY  +  2/*-  18.  625a;4  +  a;2  +  1. 


IDENTITIES  AND   THEOREMS,  99 

19.  x^  -  66:^2  _[_  625.  25.  2500:?;4  _|_  4^2  _|_  4^ 

20.  a;«+14a;*+625  (4  factors).  26.  3aV  -  51aV  +  48^2. 

21.  x^  —  ll2;3  +  X.  27.  80^;^  +  115a:''  +  405a;^ 

22.  a'l)^  -  27^2^4  _|.  j6^  2g  7^^4^4  _49a2:y+567a^/^ 

23.  80^5  -  120a3  +  5a.  29.  4^*  +  162a;8  +  2. 

24.  243:^;y  +  24^;^  +  48^1  30.  4:X^  -  "^b^x^  -  1, 

Factoring  Quadratics  by  Completing  the  Square. 

118.  Model  M.— Factor  x^  +  56^  +  768. 

x^  +  56a;  +  768  =  a;^  +  56a;  +  784  -  16 
=  (a;  +  28)2  _  42 
=  (a;  +  28  +  4)(a;  +  28-4) 
=  (a;  +  32)(a;  +  24) 

EXERCISE  LXIV. 

In  the  same  2vay  factor : 

1.  a;2  —  62a;  +  945.  11.  x^  -  200a;  —  281600. 

2.  a;2  +  68a;  +  1155.  12.  x^  +  400a;  -  422400. 

3.  a;2  -  10a;  -  704.  13.  x^  -  700a;  +  105600. 

4.  a;2  -  50a;  +  616.  14.  x^  +  800a;  +  153600. 

5.  a;2  -  12a;  -  1260.  15.  x^  -  900a;  +  201600. 

6.  a;2  -  14a;  -  1176.  16.  x^  -  800a;  +  158400. 

.  7.  a;2  -  82a;  +  1512.  17.  x^  -  800a;  -  614400.    . 

8.  a;2  -  10a;  -  875.  18    x^  +  800a;  +  134400. 

9.  a;2  -  8a;  -  768.  19.  x^  -  900a;  +  170100. 
10.  a;2  +  74a;  +  1344.  20.  x^  -  1400a;  -  633600. 

119.  This  process  becomes  somewhat  more  difficult  when 
the  coefficient  in  the  middle  term  is  an  odd  number. 

Model  N. 
a;2  +  11a;  -  726  e:  a;^  +  11a;  +  (  )  -  726  -  (  ) 
=  a;2  +  11a;  4-  30.25  -  756.25 
=  (a;+  5.5)2  _  (27.5)2 
=  {x+  5.5  +  27.5)(a;  +  5.5  -  27.5) 
=  (a;  +  33)(a;  -  22). 


100  ALGEBRA, 


EXERCl 

(SE  LXV. 

In  the  same  tvay  factor  ' 

1. 

x^  +  6x-   594. 

16. 

x^  -   lOlo;  +  2520. 

2. 

x^  -  9x  -   792. 

17. 

x^  —  57^  -  2268. 

3. 

x^  -   152;  -  1134. 

18. 

x'^  +  135:c  +  4536. 

4. 

x^  -  51x  +  648. 

19. 

x'  +  145a;  +  5184. 

5. 

x^  +  81:^  +  1458. 

20. 

x^  -  21x  -  4860. 

6. 

x^  -  76x  +  1386. 

21 

x^  -  2lx  -   1960. 

7. 

x^  —   75^  -  13356. 

22. 

x^  +  117a;  +  3240. 

8. 

x^  +  87^  +  1782. 

23. 

x^  -   119^;  +  2940. 

9. 

x^  +  67^  +  1120. 

24. 

x^  -  l^x  -  2880. 

10. 

^2  _  79^  _^  1350. 

26. 

x^  +  165:?;  +  6804. 

11. 

x^  +  SSx  +  1512. 

26. 

x^  -  Id^x  +  4800. 

12. 

^2  _  37^  _|_  1620. 

27. 

x^  +  31x  -   2520. 

13. 

^2  _  99^  _j_  2240. 

28. 

2:2  _^  139^  _^  3730^ 

14. 

x^  —  3x  —  5400. 

29. 

.2:^  -  99^;  +  1944. 

16. 

a;2  +  129:^;  +  3780. 

30. 

x^  +  9x  -   6300. 

120.  A  still  further  complication  is  introduced  when  the 
x^  term  has  a  coefficient  greater  than  1. 

Model  0.— Factor  %x^  -  II2;  -  72. 

The  most  convenient  way  to  attack  an  example  of  this 
kind,  if  it  is  not  practicable  to  factor  it  by  inspection,  is  to 
divide  the  expression  throughout  by  the  coefficient  of  x^,  so 
as  to  reduce  the  expression  to  the  form  of  those  we  have 
just  been  considering.  It  must  not  be  forgotten  that  this 
divisor  must  be  restored  to  the  other  factors  when  we  find 
them. 


2        11^ 


>---^^+(gr---(ir 


llo;       121  _  1849 

6  "^  144    Tsr 

43\2 


_/  IIV       /43\ 

=l"-r2)  -(12) 


IDENTITIES  AND  THEOREMS.  101 


+  isjl^  -  13 


-i         11 

These  factors,  multiplied  by  the  coefficient  we  divided  by 
at  first,  will  be  thq  factors  of  the  given  expression. 

Qx^  -  11a;  -  72  =  ^{x  +  %){x  -  f) 
=  ^{x+%),2{x-^) 
=  (3a;  +  8)(2ic-9). 

Model  P.  400:?;^  +  678:?^  —  135  is  somewhat  harder  to 
factor  by  inspection.     Dividing  by  400, 

678    _135^    2   I    6^8         /339y  _135        /339\^ 
^  +  400  ^      400  "^  "^  +  400  ^  +  UoO  /        400        \  400  ) 

_  ^  _^  t)  /«^  +  160000       160000  -  r  +  400/         \400y 

=  r  +  400  +  4ooy  r  +  400   400/  -  r  +  4ooy  r    400; 

Whence  the  factors  of  400:?;2  _j_  673^  _  ^35  ^^e 

=  (8:?;  +  15)(50:^-9). 

121.  Practice  in  the  application  of  this  method  of  fac- 
toring may  be  had  by  trying  some  of  the  harder  examples 
of  the  preceding  chapter.  It  is  not  well  to  spend  much 
time  on  such  examples  as  the  following,  because  an  easier 


102  ALGEBRA. 

method  will  be  given  later,  in  which  the  factors  can  be 
found  by  a  formula. 

EXERCISE  LXVI. 

Factor  the  folloioing  dy  completing  the  square : 
1.  30^2  _|_  56^  _|_  24.  ^    24:X^  +  Q7x  +  45. 

2  21x'^  -  1042:  +  60.  7.  362;^  +  81^  -  40. 

3  36.^■2  +  231^  -  345.  8.  150^^  _  175^  _  294. 
4.  200:?;2  _  9^^  _  ^35^               g    43^2  _|_  73^  _  3^5^ 

6.  210a;2  -  23x  -  72.  10.  4:00x^  -  651^;  +  108. 

122.  This  method  can  be  used  to  factor  any  *  quadratic 
expression,  even  when  the  second  number  in  the  trans- 
formed expression  is  not  a  perfect  square;  because  we  can 
find  the  approximate  square  root  of  any  number  expressed 
in  figures. 

Model  a.— Factor  3^^  _|_  g^^:  +  1. 

2    ,    8:r    ,    1        „    ,82:    ,    16        13 
^   +y  +  3^^   +-3    +^--9 


13.0000 
9 

3.605 

400 
396 

6 

40000 
36025 

7205 
5 

397500 

X    + 


3  y\    '  3      3  y 

=  {x  +  1.33  +  l,2){x  +  1.33  -  1.2) 
=  {x  +  2.53)(ic  +  .13) 

*  An  apparent  exception  occurs  wlien  tlie  transformed  expression 
becomes  the  sum  of  two  squares,  instead  of  the  difference.  Thus 
x^  —  10a;  +  29  =  {x  -  5)2  -f  (2)2,  to  which  Theorem  A  does  not  apply. 
This  difl&culty  will  be  dealt  with  later. 


IDENTITIES  AND   THE0UEM8.  103 


EXERCISE   LXVII. 

In  the  following  examples  figure  the  coefficients  of  the 
factors  to  two  places  of  decimals : 

1.  x^  -{-  %x  —  1.  16.  x^  -{-  ^x  —  2. 

2.  x^  -^  ^x-\-  2.  17.  x^  -  Ux  -  7. 

3.  x^  +  Qx  +  7.  18.  x^  +  I'lx  -  10. 

4.  x;^  —  Sx  +  14.  19.  x^  -  21a;  -  20. 

5.  x^  -  16x  +  61.  20.  x^  -  332;  —  2. 

6.  x^  —  10:^;  +  20.  21.  x^  +  x  —  1. 

7.  x^  —  20:^;  —  20.  22.  2x^  +  3:^  —  4. 

8.  x^  +  12x  +  30.  23.  3a;2  -\- "^x  —  2. 

9.  x^  -^  Sx  —  1.  24.  4a:^  +  9^  "~  ^• 

10.  x^  —  ^x  —  2.  25.  2:?;2  +  a;  —  1. 

11.  x^  +  3a;  +  1.  26.  a;2  +  3a;  +  1. 

12.  x^  -  \lx  +  6.  27.  3a;2  +  10a;  +  1. 

13.  a;2  -  7a;  +  11.  28.  bx^  +  8a;  +  2. 

14.  x^  -  15a;  +  42.  29.  2a;2  +  8a;  +  5. 

15.  a;^  +  5a;  +  3.  30.  4a;^  -\-  bx  —  ^. 


CHAPTEK  Y. 
FACTORABLE   EQUATIONS. 

123.  A  theorem  has  been  defined  as  a  general  statement 
requiring  demonstration;  and  an  axiom  as  a  general  state- 
ment not  requiring  demonstration.  One  of  the  most  im- 
portant axioms  in  elementary  algebra  is  the  following : 

124.  The  product  of  two  or  more  factors  can  never  be 
zero  unless  at  least  one  of  those  factors  is  itself  equal  to 
zero.* 

Two  Answers  to  One  Question. 

125.  Model  A. — A  square  box  7  inches  high  has  160 
square  inches  more  in  its  lateral  surface  than  on  its  bottom. 
What  is  the  size  of  the  bottom  ? 

Let  X  =  length  (and  breadth)  of  the  bottom. 

Then  Ix  —  area  each  side  and  x'^  =  area  bottom. 
Q  282;  -  x^=  160 

©  0  =  160  -  28^;  +  x^  (T)  -  28^  +  x^ 

(Z)0  =  (^x-8){x-  20)  ©  factored  f 

®  0  =  ^  -  8  from  (3)  by  Ax.  A 

®  S=^x  ®  +  8 

(6)  0  =  :?;  —  20  from  (3)  by  Ax.  A 

®  20  =  i?;  ©  +  20 

*  This  axiom  will  be  hereafter  referred  to  as  Axiom  A. 

f  When  an  equation  is  so  arranged  that  all  its  terms  are  on  one 
side  and  zero  on  the  other — in  other  words,  so  that  we  have  an 
algebraic  expression  equated  to  zero — then  the  factors  of  that  expres- 
sion are  sometimes  loosely  called  the  ''factors  of  the  equation." 

104 


FACTORABLE  EQUATIONS.  105 

126.  This  example  has  two  answers,  or,  as  we  sometimes 
say,  there  are  two  values  for  x  which  satisfy  the  equation. 
This  does  not  mean  that  the  same  quantity  x  can  be  equal 
to  two  different  numerical  expressions  at  the  same  time; 
the  box  canuot  be  8  inches  square  and  also  20  inches 
square.  But  a  box  8  inches  square  would  have  the  prop- 
erties described  in  the  equation,  and  so  would  a  box  20 
inches  square;  there  are  two  sizes  of  square  box  that  could 
be  made  so  as  to  be  7  inches  high  and  at  the  same  time  to 
have  160  square  inches  more  in  the  lateral  surface  than  on 
the  bottom. 

EXERCISE  LXVIII. 

The  following  examples  have  each  two  answers : 

1.  Both  ends  and  one  side  of  a  rectangular  field  require 
41  rods  of  fencing,  and  the  area  of  the  field  is  200  square 
rods.     Find  its  dimensions. 

2.  A  field  17  rods  long  is  made  square  by  cutting  off  72 
square  rods  from  one  end.     Find  the  width  of  the  field. 

3.  A  farmer  trades  grain  with  a  seedsman  for  a  basket 
of  new  seed,  and  agrees  to  fill  a  basket  for  him  for  every 
quart  the  basket  contains;  he  filled  the  basket  12  times, 
and  then  found  that  he  had  given  just  one  bushel  too 
much.     How  many  quarts  did  the  basket  hold  ? 

4.  A  man  contracted  to  pay  15  cents  a  foot  for  gilt 
picture-moulding  and  20  cents  per  square  yard  for  straw 
matting  used  in  furnishing  a  square  room  in  his  house. 
He  was  astonished  to  find  that  the  picture-moulding  cost 
him  $4  more  than  the  matting.  What  was  the  size  of  his 
room  ? 

5.  In  another  room,  where  there  were  2  yards  more  in 
the  length  than  in  the  breadth,  the  moulding  at  15  cents 
per  foot  cost  14.20  more  than  the  matting  at  20  cents  per 
square  yard.     What  were  the  dimensions  of  the  floor? 


106  ALGEBRA, 

6.  There  is  a  cylinder  8  inches  high,  whose  lateral  sur- 
face (2 TrrA)  exceeds  the  area  of  its  base  {nr^)  by  88  square 
inches.     Find  its  radius.     (Use  n  =  d\.) 

7.  A  square  reception-room  11  feet  high  has  walls  and 
ceiling  papered;  8  more  rolls  of  paper  (36  square  feet  to 
the  roll)  are  required  to  paper  the  walls  than  are  required 
for  the  ceiling ;  what  is  the  size  of  the  room  ? 

8.  A  man  sold  a  horse  for  $102  and  found  that  his  loss 
per  cent  was  one-eighth  of  the  number  of  dollars  he  had 
paid  for  the  horse.     How  much  had  he  paid  ? 

9.  A  stock-raiser  bought  sheep  for  1210,  but  after  he 
had  lost  5  by  sickness,  he  sold  what  he  had  left  of  the 
flock  for  1150,  a  loss  of  $1  a  head.  What  did  they  cost 
him  apiece  ? 

10.  A  man  bought  9  tons  of  coal  in  an  inland  town, 
and  sold  enough  to  get  his  money  back  at  a  profit  of 
$4  per  ton;  afterwards  the  price  rose  $6  per  ton,  and 
he  could  have  got  his  money  back  by  selling  2  tons  less. 
How  much  did  he  pay  for  his  coal  ?  How  many  tons  did 
he  sell  ? 

127.  If  we  represent  by  a  the  number  which  is  given  as 
7  in  the  last  illustrative  problem,  and  by  I?  the  number 
which  is  given  as  160,  we  can  state  five  new  problems  by 
putting  the  following  values  for  a  and  b  in  the  statement 
of  that  illustrative  problem : 


a  =  11; 

b  =  340 

a  =  8; 

b  =     60 

a  =     5; 

b  =     91 

a  =  9; 

b  =  160 

a  =    4:; 

b  =z    62 

In  the  same  way  successive  sets  of  figiires  are  given  for 
each  of  the  ten  problems  in  the  last  exercise : 

Example  1.             {a)     27         42       34  39       148 

(a  =  41;  Z>  =  200)       {b)     91       220     144  187     2400 


FACTORABLE  EQUATIONS,  107 

Example  2.  {a)     23       30       28         40         44 

(a  =  17;   Z^  =  72)  (Z>)    132     200       75       375       480 

Example  3.  {a)     13       22       11         16         15 

(a  =  12;  Z>  =  4  pecks)     (Z*)       5         5         3  3-i         4i 

Example  4.  {a)  lOcts.  llcts.  8cts.  13cts.  25cts. 

(a  =  15;  ^  =  20;  (Z>)  15cts.  16cts.  18cts.  36cts.     $1 

c  =  $4)  {c)  $1.05  $2.30  56cts.  $1.20      $2 

Example  5.  {a)    2  ft.    2  ft.  2  yds.  2  yds.  3  yds. 

(a  =  2  yds.;  Z^=15cts.;   (b)  lOcts.  7ets.  lOcts.  15cts.  llcts. 

c  =  20cts. ;  d=  $4.20)  (c)  21cts.  15cts.  18cts.  27cts.  36cts. 

(d)  $1.05  $1.08  $2.00  $3.00  $2.78 

Example  6.  (a)      25         5  6  9        13 

(a  =  8";^  =  88sq.in.)    {b)     154       66      110      176      418 

Example  7.  («)      12       10|^        9        10  6 

(a  =  11;  Z>  =  8)  {b)        7       10  5        11  3 

Example  8.  (a)    $144  $147  $105  $192  $105.60 

(a  =  $102;  ^^  =  $1)         {b)       i         i        i        iV        tV 

Example  9.  (a)    120     215      256      210      280 

(«  =  $210;  b  =  6;  (b)         d         S  2  8  3 

c  =  $150)  (c)   85  140   210   135   225 

Example  10.     (a)  10  12  50  30  32 

{a=:9;     b  =  U;            (b)  3  2  1  4  3 

c;=  $6;  d  =  2)              \c)  4  3  5  2  2 

(J)  2  2  15  3  4 

Both  Answers  Alike. 

128.  Model  B. — In  example  1  in  the  preceding  exercise 
suppose  the  room  had  been  6  feet  high,  and  that  4  more 
rolls  of  paper  were  required  for  the  walls  than  for  the  ceil- 
ing. 

Let  X  =  the  length  (and  breadth)  of  the  room. 


108  ALGEBRA, 

Then  Qx  =  area  of  each  side,  and  x^  =  area  of  ceiling. 
©  24:x  -  x^  =  144 

(2)  x^  -  24x  +  lU     =  0        ®  -  24:X  +  x^ 
®  {x  -  12)(:z;  -  12)  :=:  0       (D  factored 
®  i?;  —  12  =  0  from  ®  by  Ax.  A 

®  2;  ==  12  ®  +  12 

Here,  on  account  of  a  peculiar  selection  of  figures,  the 
two  answers  to  the  problem  are  alike.  Similar  results  are 
obtained  if,  instead  of  6  feet  high  and  4  more  rolls,  we  say 
9  feet  high  and  9  more  rolls,  or  12  feet  high  and  16  more 
rolls,  or  15  feet  high  and  25  more  rolls. 

Try  the  effect,  in  problems  1,  2,  3,  4,  and  6,  of  the 
numbers  given  below : 

Example      12  3  4  6 


(a)  60 

18 

16 

20 

rv 

(b)   450 

81 

2  bush. 

24 

154 

(c) 

16 

Answers  Apparently  Different. 

129.  If  we  find  two  different  numbers  as  the  values  of  x 
in  solving  a  quadratic  equation,  we  naturally  conclude  that 
there  are  two  answers  thus  implied  for  the  problem  which 
gave  rise  to  the  equation.  We  sometimes  find,  however, 
that  the  two  answers  thus  indicated  are  the  same. 

Model  C. — To  divide  28  into  two  parts,  whose  product 
shall  be  75. 

Let  x  =  one  part ;  then  28  —  a;  =  the  other. 

Q  x{2S  —  x)  =  76  (product  of  both  parts) 

©  282;  —  x^    =75  same  as  (T) 

(D  :c2  -  28ir  +  75  =  0      ®  +  x^  -  28a; 

®  {x-26){x-d)  =  0      ©factored 

®  x  -  25  =  0 

(6):z;  =  25 

(T)x  -  3  =  0 

@x  =  3 


FACTORABLE  EQUATIONS,  109 

Here  the  first  answer  means  that  one  part  can  be  25; 
then  the  other  must  be  3 ;  or  that  one  part  can  be  3 ;  then 
the  other  must  be  25.  The  two  answers  are  therefore 
really  the  same. 

EXERCISE    LXIX. 

1.  A  rectangular  field  containing  one  acre*  requires 
924  feet  of  fencing.     What  are  its  dimensions  ? 

2.  Two  cubical  bins,  side  by  side,  extend  the  whole 
length  of  a  16 -foot  wall,  and  contain  9|^  cords  *  of  kindling- 
wood.     What  is  the  size  of  each  ? 

3.  The  number  30551  has  two  factors  whose  sum  is  360. 
What  are  they  ? 

4.  A  rectangular  room  requires  68  feet  of  picture-mould- 
ing, which  runs  above  the  top  of  the  windows;  and  the 
same  room  requires  32  yards  of  carpet.     What  is  its  size  ? 

5.  A  similar  room  requires  20  yards  of  border  f  half  a 
yard  wide,  and  18  yards  of  carpet  one  yard  wide.  What  is 
the  size  of  the  room  ? 

6.  A  wharf  which  projects  27  feet  into  the  water  is  made 
of  two  square  platforms,  and  its  total  area  is  377  square 
feet.     What  is  its  greatest  width,  and  its  least  ? 

7.  Forty-six  rods  of  fencing  are  required  for  a  field  whose 
diagonal  is  17  rods.     What  is  the  size  of  the  field  ? 

8.  The  government  of  Utopia  once  enacted  that  each 
liquor  saloon  should  pay  $10  for  its  own  license,  and  the 
same  amount  for  every  other  license  granted  in  the  same 
ward,  and  at  the  same  time  refused  to  grant  licenses  out- 
side of  the  two  central  wards.  Thus  48  licenses  were 
granted,  yielding  a  license  tax  of  $12,800.  How  many 
licenses  in  each  ward  ? 

*  An  acre  is  10  square  chains,  and  a  chain  is  66  feet  long  ;  a  cord 
of  wood  measures  4  X  4  X  8  ft. 
t  Not  including  mitres. 


110  ALGEBRA. 

9.  Two  years  later  the  number  of  licenses  had  increased 
to  208,  and  the  total  tax  had  become  1217,040.  How 
many  then  in  each  ward  ? 

10.  A  rectangular  corner  lot  and  two  square  lots  adjoin- 
ing contain  in  all  9972  square  feet,  and  have  altogether  114 
feet  frontage  on  each  street.  All  the  lots  on  a  street  have 
the  same  depth.  What  are  the  dimensions  of  the 
lots? 

11.  Two  numbers  are  reciprocals  and  their  sum  is  2y\. 
What  are  they  ? 

12.  Two  men  are  separately  hired  to  lay  the  curbstone 
on  opposite  sides  of  the  same  street,  and  receive  for  the 
whole  job  8  days^  pay  between  them;  one  third  of  the  job 
was  accomplished  the  first  day.  How  many  days  did  it 
take  each  man  to  do  his  own  work  ? 

13.  By  one  pipe  a  tank  is  filled,  and  then  by  another 
immediately  emptied,  the  whole  operation  requiring  an 
hour  all  but  10  minutes;  then  both  pipes  are  used  to  fill 
the  tank  again,  which  is  done  in  12  minutes.  How  long 
for  each  pipe  ? 

Meaning  of  Negative  Answers. 

130.  Sometimes  one  answer  is  negative  and  still  can  be 
interpreted  as  a  reasonable  answer.  It  is  generally  neces- 
sary to  change  the  sense  of  some  word,  so  as  to  have  it 
mean  just  the  opposite  of  its  meaning  in  the  statement  of 
the  problem. 

Model  D. — Two  laborers  applied  for  work  on  a  farm, 
and  were  each  sent  to  build  a  rod  of  fence,  working  by  the 
hour.  After  they  were  through,  the  overseer  decided  that 
one  of  the  men  could  build  3  rods  a  day  more  than  the 
other,  and  that  between  them  they  had  earned  half  a  day's 
pay.     How  many  rods  could  each  build  per  day  ? 


PAGTOHABLE  JSQUATlOm.  Ill 

Let  X  =  the  number  of  rods  the  poorer  laborer  builds 
in  a  day;  then  a;  +  3  =  the  number  of  rods  the  better 
laborer  builds  in  a  day.  , 

^  x'^  x  +  d      2 

(2)  2x  +  6  +  2x  =  x^  +  '^x  ®   X  2x{x  +  3) 

@x^  —  X  —  6  =  0  ©—  4a;  —  6 

®  (^  -  3)(i^;  +  2)  =  0  (D  factored 

Whence  we  obtain  x  =  3  and  :?;-j-3  =  6;  i^==— 2  and 
r?;+3=:l. 

The  answer  a;  =  —  2  becomes  intelligible  if  we  assume 
that  to  build  —  2  rods  means  to  destroy  2  rods,  and  tliat 
the  destructive  laborer  has  to  pay  by  the  hour  for  his  amuse- 
ment. While  the  better  laborer  is  building  his  rod  and 
earning  thereby  a  day^s  pay,  the  other  fellow  gets  through 
with  destroying  his  rod  and  has  to  pay  for  the  half -day  that 
he  uses  up  in  so  doing. 

If  we  represent  by  a  the  number  3  in  the  above  problem, 
and  by  b  the  number  ^ ,  we  can  substitute  the  following 
numbers  for  a  and  &,  and  obtain  in  each  case  a  new 
problem. 


(«) 

5         7         9         33 

39 

(*) 

i        -h       io         i^ 

EXERCISE    LXX. 

jV 

1.  A  field  20  rods  by  18  is  made  352  square  rods  in  area 
by  cutting  off  a  strip  of  uniform  width  on  the  side,  and 
adding  to  it  a  strip  of  the  same  width  on  the  end.  What 
is  the  width  of  the  strips  ? 

2.  If  a  fraction  whose  numerator  is  70  be  inverted  and 
multiplied  by  6  its  new  value  will  exceed  its  original  value 
by  1.     Find  the  fraction. 


112  ALGEBRA, 

3.  A  boat  which  is  able  to  go  5  miles  an  hour  goes  21 
miles  southward  in  a  certain  river  and  then  24  miles  north- 
ward, the  whole  time  spent  in  travelling  being  11  hours. 
How  fast  does  the  stream  flow  southward  ? 

4.  Both  pipes  at  a  certain  water-tank  are  opened,  and 
the  tank,  being  empty  at  noon,  is  filled  at  2  o'clock  ;  but 
if  the  pipes  were  opened  separately,  one  each  day,  the 
tank  would  be  filled  3  hours  later  by  one  than  by  the 
other.     At  what  time  is  the  tank  full  in  each  case  ? 

5.  A  manufacturer  employs  his  son  as  superintendent  at 
$3  per  day,  with  the  understanding  that  the  average  amount 
saved  by  him  in  the  daily  expenses  of  the  factory  is  to  be 
added  to  his  daily  wages.  When  the  balance  due  to  his 
savings  has  reached  $400  on  the  books,  he  finds  that  the 
concern  owes  him,  including  his  wages,  five  times  the 
average  daily  saving  of  the  factory.  What  is  the  average 
daily  saving  ? 

For  these  examples  the  following  numbers  will  serve  to 
state  oieiu  problems : 

Example  1.    {a)     20   16   25   32   35 

(a  =  20;  ^  =  18;  {b)     13   10   22   30   12 

c  =  352)       {c)    230  105  540  880  342 


Example  2.        (a) 

8 

72 

16 

45 

80 

(a  =  70;  &  =  6;      {b) 

4 

36 

144 

900 

16 

c  =  1)                     {c) 

3 

5 

7 

11 

15 

Example  3.        (a) 

3 

7 

6 

13 

8 

{a  =  5;  &  =  21;     {b) 

12 

34 

30 

62 

22 

c  =  24:;  d=ll)   (c) 

10 

33 

32 

63 

30 

(d) 

8 

10 

19 

10 

8 

Example  4.        {a) 

3 

6 

4 

6 

8 

(a  =  2;  6  =  3)        (b) 

8 

5 

15 

35 

63 

FAGTORABLE  EQUATIONS,  113 


Example  5.   [a) 

4 

5    2    6   10 

{a  =  ^;b  =  ir,      W 

T^ 

i        i        H      6^ 

c  =  400;  d  =  6)    (c) 

80 

900  1152  3840  2500 

(d) 

8 

90   10    6   13 

Answers  Suggesting  Related  Problenns. 

131.  A  negative  answer  to  a  quadratic  equation  may  not 
be  in terpre table  as  a  reasonable  answer  to  the  problem 
which  gave  rise  to  the  equation,  but  may  suggest  a  new 
problem,  closely  related  to  the  given  one,  and  having  the 
negative  and  the  positive  answers  interchanged.  Such  is 
the  following  : 

Model  E. — A  farmer  bought  a  herd  of  cows  for  1400;  if 
there  had  been  4  more  in  the  herd,  the  price  would  have 
been  15  less  apiece.     How  many  were  there  ? 

Let  X  =  the  number  of  cows  in  the  herd. 

^400         400     _ 

©  400^;  +  IGOO  =  400a;  +  hx^  +  20a;  ®^x{x  ^  4) 

(3)  hx^  +  20a;  -  1600  =  0  ©-1600-400a; 

©a;2  + 4a;- 320  =  0  ®-^5. 
(5)  a;  =  16;  a;  =  —  20. 

Now  we  can  change  the  algebraic  sense  of  the  problem 
by  changing  words  like  ^^more'^  to  ''less  "  and  vice  versa. 
The  new  problem  would  read  : 

A  farmer  bought  a  herd  of  cows  for  $400;  if  there  had 
been  4  less  in  the  herd  they  would  have  cost  $5  more 
apiece.     How  many  were  there  ? 

Let  X  =  the  number  of  cows  in  the  herd. 

^  400         400 

^^     a;        a;  —  4 
Whence  a;  =  20  or  —  16. 


114  ALOEBBA. 


EXERCISE    LXXI. 

Of  ihe  following  'problems,  the  negative  anstvers  suggest 
related  problems  in  the  opposite  algebraic  sense  : 

1.  There  are  40  rooms  in  a  house,  and  3  more  rooms  on 
a  floor  than  there  are  floors.     How  many  floors  ? 

2.  Comparing  a  pail  with  a  140-quart  tub,  a  man  con- 
cluded that  the  tub  held  as  many  pailfuls  as  the  pail  held 
quarts.  He  found,  however,  that  the  tub  held  4  pailfuls 
more.     What  was  the  capacity  of  the  pail  ? 

3.  In  order  to  save  4  hours  on  his  regular  trip  of  126 
miles,  a  stage-driver  finds  it  necessary  to  go  2  miles  an  hour 
faster  than  his  usual  rate.     What  is  his  usual  rate  ? 

4.  What  is  the  price  of  eggs  when  10  more  in  a  dollar^s 
worth  lowers  the  price  4  cents  a  dozen  ? 

5.  A  rectangular  field  contains  1^  acres;  if  its  length  is 
increased  by  20  feet  and  its  breadth  diminished  by  18  feet 
its  area  will  be  diminished  2340  square  feet.  What  are 
its  dimensions  ? 

For  these  examples  the  following  numbers  will  serve  to 
state  new  problems  : 


Example  1. 

{a) 

35 

50 

60 

44 

112 

{a  =  40;  Z>  =  3) 

(^) 

2 

5 

4 

7 

6 

Example  2. 

{a) 

88 

176 

99 

90 

180 

{a  =  140;  Z>  =  4) 

(P) 

3 

5 

2 

H 

3 

Example  3. 

{a) 

6 

12 

4 

2 

5 

(a  =  4;  Z>  =  126; 

(^) 

176 

135 

120 

180 

150 

0=2) 

{c) 

3 

4 

li 

1 

n 

Example  4. 

{a) 

5 

2 

25 

50 

20 

{a=  10;  J  =  4) 

(^) 

1 

1 

8 

2 

5 

FACTORABLE  EQUATIONS,  116 


Example  5.     (a)    3 

3 

1 

4 

n 

{a  =  1|;  5  =  20  [V)       34 

14 

12 

-60 

-45 

c  =  18;  cZ=3340)(c)   20 

30 

10 

-16 

-30 

id)   3120 

7680 

780 

6240 

-3600 

Meaningless  Answers. 

132.  In  each  of  the  following  problems  an  answer  will 
be  obtained  which  satisfies  the  quadratic  equation,  but  has 
no  reasonable  interpretation  in  the  problem  which  gave 
rise  to  that  equation ;  and  the  answer  may  not  even  suggest 
a  related  problem. 

EXERCISE    LXXII. 

1.  A  square  box  7  inches  high  and  without  a  cover  has 
288  square  inches  of  inside  surface.  What  is  the  size  of 
the  bottom  ? 

2.  Of  two  large  wheels,  one  makes  48  turns  more  than 
the  other  in  rolling  a  mile ;  their  tires,  straightened  out  and 
laid  end  to  end,  would  reach  21  feet.  What  is  the  circum- 
ference of  each  ? 

3.  Two  sprinters  start  opposite  ways  from  the  middle 
post  of  a  straightaway  440-yard  track;  in  one  second  they 
are  21  yards  apart  and  one  of  them  finishes  his  220  yards  2 
seconds  ahead  of  the  other.     What  is  the  speed  of  each  ? 

4.  A  cistern  containing  4  gallons  is  filled  by  one  pipe 
and  then  immediately  emptied  by  another,  both  operations 
requiring  just  5  hours.  One  pipe  delivers  3  gallons  per 
hour  more  than  the  other.  How  many  gallons  from  each 
pipe  alone  ? 

5.  A  man  rows  5  miles  down-stream  and  back  in  4  hours. 
The  stream  flows  3  miles  an  hour.  How  fast  can  the  man 
row  in  still  water  ? 


116  ALGEBRA, 

For  these  examples  the  folloioing  numbers  will  serve  to 
state  new  problems : 


Problem  1. 

{a) 

8 

10 

15 

20 

3 

{a  =  '7;  b  =  288) 

(^) 

185 

176 

544 

249 

36i 

Problem  2. 

{a) 

40 

10 

88 

96 

88 

{a  =  48;  J  =  21) 

(^) 

23 

32J 

22 

24f 

27' 

Problem  3. 

(a) 

400 

300 

1000 

900 

500 

(a  =  440;  h  =  21; 

(^) 

18 

22 

201 

20i 

241 

0  =  2) 

(0) 

5 

H 

22^ 

10 

t 

Problem  4. 

{a) 

6 

8 

10 

4 

21 

(«  =  4;  &  =  5; 

W 

5 

7 

3 

3 

13i 

c  =  3) 

{0) 

7 

9 

7 

5 

5 

Problem  5. 

(a) 

5 

24 

30 

25^ 

lOJ 

(a  =  5;  &  =  4; 

(^) 

9 

14 

12 

10 

5 

c  =  3) 

(0) 

4 

5 

6 

7 

2 

133.  The  answers  to  a  quadratic  equation  are  not  always 
commensurable;  in  that  case  they  can  be  approximately 
stated  as  decimal  fractions. 
Model  F. 
(Tjx^  +  dx  =  11 
©  :^2  _|_  3^  _  11  ^  0  ®  -  11 

(D  ^2  _|_  3^  _j_  9  _  5_3  ~  0  same  as  © 

®  (^  +  1)^  -  (3.64)2  ^  0  same  as  ® 

®  (^  +  I  +  3.64)(:c  +  I  _  3.64)  =  0    ®  factored 
(D  {x  +  6,l^){x  -  2.14)  =  0        same  as  ® 

X  =  2.14     or 
X  =  -  5.14 
The  following  solutions  are  worthy  of  attention : 
Model  G. 

®  a;2  =  16 

©  a;2  -  16  =  0  ®  -  16 

®  {x  +  4:){x  —  4)  =  0  ®  factored 

®a;  +  4  =  0;^=-4) 

®  ^  -  4  =  0;  ic  =  4        [    ®  Ax.  A 


Ans, 


FACTORABLE  EQUATIONS.  117 

Hence  there  are  two  square  roots  to  16  (or  to  any  arith- 
metical number);  one  is  +  and  the  other  — . 
Model  H. 

®x^  -  'dx  =  0  ©  -  3a; 

®x{x  —  ^)  =  0  (2)  factored 

Notice  that  the  value  a;  =  0  satisfies  the  equation. 

EXERCISE  LXXIII. 

1.  What  number  divided  into  81  will  give  itself  for  a 
quotient  ? 

2.  Three  times  the  square  of  a  number  is  equal  to  10 
times  the  number  itself.     What  is  the  number  ? 

3.  A  room  is  twice  as  long  as  it  is  wide,  and  its  floor  con- 
tains 144^  square  feet.     How  wide  is  it  ? 

4.  A  square  room  9  feet  high  takes  twice  as  much  paper 
to  cover  the  walls  as  to  cover  the  ceiling.  What  is  the  size 
of  the  ceiling  ? 

5.  In  a  current  of  2  miles  an  hour  a  man  takes  8  hours 
longer  to  row  24  miles  up-stream  than  to  row  the  same 
distance  down.     How  fast  can  he  row  in  still  water  ? 

6.  A  field  20  rods  by  18  is  made  353  square  rods  in  area 
by  cutting  off  a  strip  of  uniform  width  on  the  side  and 
adding  a  strip  of  the  same  width  on  the  end.  What  is  the 
width  of  the  strips  ? 

7.  A  square  box  9  inches  high  has  300  square  inches 
more  on  the  sides  than  on  the  bottom.  What  are  the  di- 
mensions of  the  box  ? 

8.  One  man  takes  4  hours  longer  than  another  to  saw  a 
cord  of  wood,  and  both,  working  together,  can  saw  it  in  3 
hours.     How  long  for  each  ? 

9.  Two  wheels,  the  sum  of  whose  circumferences  is  equal 
to  8  yards,  roll  the  same  distance;  and  the  number  of  turns 


118  ALGEBRA. 

made  by  one  wheel,  added  to  the  number  of  turns  made  by 
the  other,  gives  a  sum  numerically  equal  to  the  number  of 
yards  traversed.*  Find  the  circumferences  of  the  wheels. 
10.  A  line  100  centimetres  long  is  divided  into  two 
parts,  such  that  the  longer  part  is  contained  in  the  whole 
line  as  many  times  as  the  shorter  part  is  contained  in  the 
longer  part.     What  is  the  length  of  each  part  ? 

A  New  Kind  of  Number. 

134.  The  solution  of  certain  quadratics,  of  which  the 
simplest  \^  x^  =  —  1  or  x^  -\-  \  —  0,  leads  to  an  expression 
which  has  no  interpretation  according  to  the  rules  of 
algebra,  so  far  as  we  have  studied  them  at  present.  This 
expression  is   V—  1,  and  it  arises  as  follows: 

Model  I. 

Q  a;2  =  -  1 

®  ^-'^+1  =  0  ® +  1 

@  x'^  —  {—  1)  =  0  @  in  form  of  x^  —  y^ 

0  ^2  _  (  i/ZTiy  =  0 

(5)  (^x  +  V~l){x-  V~l)=  0         ®  factored 

®  ^  +  y— 1  =  0;  X  =  -  ^^H  ^  Ax    A 

When  we  remember  that  two  factors  must  be  exactly 
alike  in  order  that  their  product  may  be  a  perfect  square, 
and  also  that  two  numbers  which  are  alike  in  respect  to 
sign  can  never  have  a  negative  product,  we  see  that  it  is 
impossible  for  a  negative  quantity  to  have  a  square  root 
like  any  other  algebraic  expression  such  as  we  have  met 
hitherto.  It  is  customary  to  represent  this  new  and  strange 
expression,  V—  1,  by  the  small  letter  i,  so  that  P  =  ■—  1. 

*  Have  a  letter  for  the  number  of  yards  traversed ;  also  one  other 
letter. 


FACTORABLE  EQUATIONS. 


119 


Then  i^  is  a  part  of  our  algebraic  system,  and  while  we  can- 
not say  that  i  is,  we  will  assume  that  it  can  be  dealt  with 
by  ordinary  algebraic  laws. 

Expressions  in  which  i  appears  in  the  first  degree  are 
called  imaginary. 

Model  I. — The  solution  of  the  equation  o;'^  =  —  1  now 
becomes : 


®  a;^  =  v' 

(D  x"  -{'=0 

(i)-v> 

®  {x  +  i){x  -i)  =  0 

©  factored 

(J)  X  +  i  =  0;    x  =  —  i^ 
®  x-i-0;    x  —  i       1 

(D  Ax.  A 

0 


Model  J. 

@  ^2  -  16|2  zz:  0      Subst.  i^ 
@  {x  +  4:i){x  —  4:i)  =  0 
0  a;  -f  4^  =  0 ;    .T  =  —  4i  I 

@    X  —    4:1   =    0;      X   =   4:1  ) 


-lin® 
(D  factored 

(D  Ax.  A 


Model  K. 

(i)x'^+6x  +  13  =  0 
(2)x^-\-6x-\-9-{-4  =  0         same  as  ® 
®x^+6x  +  9  —  4f'^  =  0      subst.  *^  =  —  1  in  @ 
(A)  {x  +  d  +  2i){x  +  3~  2^)  =  0   (3)  factored 
®  x+3+2i=z0;    x=:-3-2i 
®x+d-2i=0;   x=-d  +  2i_ 


®  Ax.  A 


135.  The  great  advantage  of  the  substitution  of  i^  at  this 
stage  in  our  study  is  to  enable  us  to  change  the  sum  of  two 
squares  to  the  form  of  the  difference  of  two  squares,  and 
thus  to  obtain  a  complete  rORM  of  solution  to  any  quad- 
ratic equation. 


120  ALGEBRA, 

EXERCISE  LXXIV. 

Solve  tlie following  equations: 

1.  —x^=^  9.  6.  ^x^  -\-x^  —  x^-^x. 

2.  ^x  —  x^  —  13.  7.  X  —  \h  — 


5  —  X 


3.  20a;  =  x^+  121.  S.  x  +  90i  =  ^(6  -  x). 

3    ,           ^,  1        16  -  :r      10  -  ^ 

X    '  Ax           X               4 

2x-7  18 

6.   =  ri;.  10. 


-  2    ' 
i^i?i^  algebraic  solutions  to  the  following  proUems : 

11.  Two  numbers  are  reciprocals  and  their,  sum  is  -J. 
What  are  the  numbers  ? 

12.  Of  two  equal  fractions  the  first  has  a  numerator  5 ; 
its  denominator  exceeds  the  numerator  of  the  second  frac- 
tion by  7,  and  exceeds  the  denominator  by  9.  Find  the 
two  fractions. 

13.  What  is  the  length  of  a  rectangular  garden,  29  square 
rods  in  area,  which  is  enclosed  by  20  rods  of  fencing  ? 

14.  In  building  a  wall  each  laborer  employed  worked  as 
many  days  as  there  were  men  on  the  job.  If  there  had 
been  10  men  less,  the  job  would  have  taken  20  days.  How 
many  laborers  were  there  ? 

15.  A  rectangular  board  12  inches  long  can  be  made 
square  by  cutting  45  square  inches  off  the  end.  How  wide 
is  it? 


FACTOBABLE  EQUATIONS, 


121 


The  Quadratic  Formula. 

136.  Any  quadratic  equation,  if  simplified  as  much  as 
possible  and  so  transposed  as  to  give  an  expression  equal  to 
zero,  will  be  in  the  form  of 

ax"^  -\-  hx  -\-  c  =  Q, 

which  is  called  the  standard  form  of  the  quadratic  equa- 
tion. 

137.  The  letters  a,  h,  and  c  are  the  three  coefficients  of 
the  quadratic,  and  of  course  for  any  given  set  of  values  for 
these  there  will  be  two  and  only  two  answers.  Thus  for 
the  following  ten  equations  the  values  of  a,  J,  and  c  are  as 
given,  and  the  answers  may  be  found  to  be  as  in  the  table. 


x^  —  bx^^  =  0 
2^2  -_  3ic  _  2  =  0 
2:?;^  —  5a;  +  2  =:  0 
6^2  —  5^;  —  6  =  0 
x^  +  10:z;  +  23  =  0 
^2  _  81  =  0 
x^  +  81  =  0 
92;2  -  16  =  0 
%x^  +  l^x  =  0 
x^  +  2a;  +  10  =  0 


a 
1 

I 

c 

-  5 

6 

2 

-3 

-2 

2 

-  5 

2 

6 

-  5 

-  6 

1 

10 

23 

1 

0 

-  81 

1 

0 

81 

9 

0 

-  16 

9 

16 

0 

1 

2 

10 

Answers. 


3;        3 

2;      i 


3  . 

2? 


9i 


—  3.586; 

9; 

9i; 

0;  -V- 

l  +  3i;  -1 


6.414 


■3i 


p        138.   It  will  be  shown  now  that  these  two  answers  to 
every  quadratic  may  be  obtained  by  the  formulae 


X  = 


—  J  +  Vb^^  —  4:ac  . 
2a  ' 


X  = 


2a 


4:ac, 


122  ALOEBRA. 

these  are  usually  written,  to  save  space,  in  one  formula, 
with  the  so-called  ambiguous  sign,  which  can  be  read  either 
plus  or  minus,  each  reading  giving  one  of  the  two  correct 
answers. 


_  -1)±  VU^  -  4:ac 
""-■  2a 

139.    This   is    called   the   ftuadratic   Formula,    and    is 
obtained  as  follows : 

Q  ax^-{-  bx  -{-  c  =  0 

(2)  x^  +  -X  +  -  =0  ®-^a 

^       '^  a         \2a)        a      ^d^  the  square 

®  l^  +  2-J    "l^^)  =  ^         sameas® 


®  factored 


^^      ^  2a  2a  2a 


®  ^+2^ 2^  =  '-'  "= 2« 


®  Ax.  A 


Whence  the  formula  x  = 

2a 


FACTORABLE  EQUATIONS.  123 

Solving  Quadratics  by  the  Formula. 

140.  Model  L. 

0  2iz:2  ^  3  -  3^ 

©  2^;''^  +  3a;  -  3  =  0         Q  +  3^  ~  3 

a  =  2  Z>^  =  9 

Z>  =  3        4(^c  =  -  24 

c  =  -  3    1)^  -  4:ac  =  33 


VU'- 

4:ac  =  5.745 

X  =  - 

-  3  ±  5.745 
4 

_  2 

.745              - 
4         '' 

8.745 
4 

= 

.686     or      - 

2.186 

Model  M. 

©I-"- 

-360:^ 

=  604:X  +  959 

©  360  -  360^;'^  =  504:?;''^  +  959:?;        ©   X  ^ 

®  864^;^  +  959a;  -  360  =  0  ©  +  360a;'^  -  360:c 


a  = 

864              P  =  919681 

864 

h  = 

959         4ac  =  -  1244160 

360 

c  = 

-  360     P  -  4.ac  =  2163841 

5184 

2592 

VP  -  4:ac  =  1471 

311040 

1244160 

-  959  ±  1471 

^  -           1728 

512 

^=1728    ''    "=^ 

2430 

1728' 

which  reduce  to 

8 
x^^        ov    x=- 

45 
32 

124  ALOEBRA. 

EXERCISE  LXXV. 

By  means  of  the  Quadratic  Formula,  find  the  Uvo  roots  * 
to  each  of  the  following  equations  : 

(Commensurable  roots.)  (Incommensurable  roots.) 

1.  Sx^  -  ^^x  +  135  =  0.     11.   "Ix^  -  10a;  -  9  =  0. 

2.  362;2  -  133:^:  =  120. 

3.  72^2  _|_  161^  _  200  ==  0. 

4.  54:^2  _  345^  ^  500  r=  0. 

5.  180a;2  +  351a;  =  50. 

6.  300^'^  -  385a;+73i  =  0. 

7.  480:^'^  -  6822;+225  =  0. 

8.  128:?;2  -  24^;  =  495. 

9.  320a;''^-1114a;+675=0. 
10.   1056:^2  =  1610:?;+1125. 

141.  The  two  roots  of  a  quadratic  equation  are  some- 
times represented  by  a  and  (i,  that  is, 


12. 

\Zx^  ^^x  ^  10. 

13. 

"IXx'  =  22x  -  23. 

14. 

lOOo;-^  =  25:c  +  36. 

15. 

x2  +  3  =  2:^;  +  1. 

16. 

x^  ^  x-\-\. 

17. 

10x2  -  IOO2;  +  203  -  0. 

18. 

64.^2  -  33x  =  100. 

19. 

202:2  r=  30:^:  +  41. 

20. 

100^2  _  looo;  -  101  =  0. 

«  =  ta '     ^  =  %a • 

EXERCISE   LXXVI. 

1.  Prove  that  a  -\-  B  — . 

a 

2.  Prove  that  a^  —  -. 

a 

3.  By  means  of  examples  1  and  2  show  that 

a(x  —  a){x  —  §)  —  ax^  -\- Ix  -\-  c, 

4.  A  quadratic  equation  whose  first  term  is  ^x^  has  for 
answers  f  and  —  | ;  what  is  the  equation  ? 

5.  In  the  equation  '^x^  -|-  4x  =  3,  find  a  —  p. 

*  The  numerical  value  of  x  which  satisfies  an  algebraic  equation  is 
called  a  ROOT  OF  the  equation. 


FACTORABLE  EQUATIONS.  125 

Factoring  Equations  of  Higher  Degrees. 

142.   Sometimes,  but  by  no  means  always,  equations  of 
degree  higher  than  the  second  can  be  factored ;  and  in  that 
case  the  factors  lead  to  solutions  by  the  same  theory  as 
ordinary  quadratics. 
Model  N. 

©  :c3  =  27 

@  2;3  -  27  =  0  ®   -  ^'^ 

®  {x-d){x^  +  ^x  +  ^)  -0     (D  factored 
®x  —  ^^0]x  =  ?^  (D  Ax.  A 

(5)  :^;2  _|_  3^  _|_  9  ^  0  (3)  Ax.  A 

(§)  X  =  — ^ — ^- —  ®  Quad.  Form. 

Ans,  X  =  3;   x  —  31 ^ 1;    x  —  3V ~  — 

These  are  the  three  cube  roots  of  27. 
Model  0. 

(Q  x^  -  Ux^  +  40  =  0 

@  {x'  -  4:){x^  -  10)  =  0 

®  {x  +  2){x  -  2){x  +  Vl6){x  -  |/10)  =  0 

(i)  X  +  2  =  0;  X  =  ~  2 

(5)  X  -  2  =  0;  X  =  2 

®  X  +  VTO  =  0;     X  =  -  |/10 

Q)  X  -  VIO  =  0;     X  =  l^io 

EXERCISE   LXXVII. 

1.  Find  the  3  cube  roots  of  64. 

2.  Find  the  4  fourth  roots  of  81. 
8.  Find  the  6  sixth  roots  of  64. 
4.  x^  -  34a;2  +  225  =  0. 
6.  36a;4  =  17x^  +  144. 


126  ALOEBRA. 

6.  x^  -  Ix^  r=  18:^;. 

7.  {x^  -  9)  (3:^-2  _|_  8a;  -  3)  r=  0. 

8.  {x^  -  3^;  +  l)(.^'^  -  3^;  -  2)  =  0. 

9.  (2;2  -  ^x)(x^  -  bx  -  10)  =  4(2;2  -  2^;). 
10.  3:^2  +  2ic  -  5  =  3a;2  +  2a;^  -  5a;. 

Factoring  by  Parts. 

143.  This  is  an  extension  of  distributive  factoring,  and 
sometimes  serves  as  an  elementary  method  for  the  solution 
of  equations  of  higher  degrees. 

The  factors  of  Mx^  —  3Mxy  +  6My^  are 

M{x^  —  3xy  +  5y^), 

This  conclusion  would  be  true  whatever  the  size  and 
shape  of  the  letter  M,  and  even  if  the  letter  M  repre- 
sented a  different  algebraic  expression  altogether.  Thus 
the  factors  of  {da  -  7d)x^  -  3(3a  -  7b)xy  +  6{3a  -  7b)  f 

are  {3a  —  7b){x^  —  3xy  +  5?/^). 

Here  the  letter  Jfis  replaced  by  the  expression  (3^^— 7^). 
Now  if  in  the  multiplication  (3a  —  7b)(x^  —  3xy  +  ^y^) 
the  second  factor  is  taken  as  the  multiplier,  the  three  ex- 
pressions (3a  —  lb)x^,  —  3(3a  —  7b)xy,  and  5(3a  —  7b)y^ 
are  the  partial  products;  distributive  factors  removed  from 
them  leave  the  multiplier,  which  is  seen  to  be  the  same  in 
all  of  the  partial  products. 

Model  P. — In  the  expression 

01^  —  5x^  —  9x  -\-  4:5 

the  first  two  terms  and  the  last  two  terms  may  be  taken  as 
partial  products; 

a^-bx^  +  {-  dx+U)=x^{x-  5)  +  9(-  x  +  5). 


FACTORABLE  EQUATIONS,  12 Y 

If  the  sign  before  the  second  partial  product  is  changed 
the  multiplicand  will  appear  the  same  in  each  partial 
product  : 

x\x  -  5)  +  9(-  :?;  +  5)  =  x\x  -  5)  -  9(^  -  5) 
=  (x-  5)(>'2-  9). 

The  same  expression  may  be  factored  by  grouping  the 
first  term  with  the  third,  and  the  second  with  the  fourth, 
to  form  the  two  partial  products : 

x^  —  bx^  —  ^x  -\-  4:^  ~  x^  —  ^x  —  bx^  +  45 
=  x{x?  -  9)  -  b(x^  -  9) 
~(x^  -^)(x  -  5). 

Of  course  the  prime  factors  of  this  expression  would  be 

(x  +  Z){x  -  3) (a;  -  5). 

EXERCISE  LXXVin. 

Factor  ly  parts  :  * 

1.  ax  +  by  —  ay  —  Ix.  6.  ^'^  —  3a:^  —  2hx  +  75. 

2.  2x-]-  xy  —  %y  -  x^.  7.  x^  +  10^:'^  -  8I2;  -  810. 

3.  x^  —  \^  -\-  ax  —  Aa.  8.   112  +  16a:  —  7x^  —  x^, 

4.  ax  +  4:X^  +  4a  +  16x.         9.   99  -  9^:  —  11^'-^  +  x\ 

5.  x^+6x  +  6-x^-  2x\  10.  121  +  I2I2;  -  x^  -  x\ 

11.  {2x  -  y){ix^  +  2xy  +  y^)  -  {2xy  +  y'){2x  -y). 

12.  (1  -  2^)(1  +  2y  +  4.y^)  -  (1  -  2^)1 

13.  y^l  -  4^  4-  4/)  -  (1  -  2yY. 

14.  a^(6  -  a)  -  (36  ~  a'). 

15.  {lOy  -  1)100/  +  (lOOy^  -  Wy)  +  {lOy  -  1). 

16.  {a+h+cY  +  Sa  +  3b  +  3c. 

17.  {x  -  17){x  +  7)  +  x{x^  -  49). 

18.  a^  -\-  ab  -j-  ac  +  be. 

19.  ace  +  %  +  ^^  +  ^y- 

*  It  may  be  necessary  in  some  cases  to  rearrange  the  terms. 


128  ALGEBBA. 

20.  6a  +  &2  _j_  GZ>  +  d^  +  ^al). 

21.  ax  ~\-  J)z  —  (bx  -\-  az), 

22.  6ax  -\-  hy  —  {hay  +  hx), 

23.  pr  —  qs  -\-  qr  —  ps. 

24.  px^  —  ag^  —  {ap  —  g'^x^), 
26.  ."^^^^  +  x^  —  {y^  —  x^). 

26.  x^  +  X  —  2y{2y  +  1). 

27.  a^x  +  «^y  +  ^^^  +  abx. 

28.  2«a;  +  3^y  —  (2^y  +  3Z>:^). 

29.  ic^  —  3?/^  +  {3xY^  —  a;''^^). 

30.  3fl^iz;^  +  (^  +  .V)(^  ~~  y)  ~  '^axy. 

31.  «^  +  32;''^^(a;  —  5?/)  —  hay, 

32.  a;y  —  9  +  ^'^  +  3?/. 

33.  ab  +  a''^^'^  — -  (x^  +  ^0- 

34.  d^  +  dx  -{-  a  —  9x\ 

35.  /^y  +  (^  +  '^)'^  +  %• 

36.  a^'(^'''  -  a^)  +  3aZ^(aZ>  -  A''^)  -  ¥{P  -  a^). 

Model  ft. — Solve  the  equation: 

Qx'  -  6x^  +  S  =^  x^  -  4:X 

@x'  -  6x^  +  8  -  x^  +  ix  :=  0     (T)  -  x^  +  4:X 
(3)  {x^  -  2){x^  -  4)  -  x{x^  -  4)  =  0 
®  (x^  -  ^){x^  -  2  -  ^)  =  0 
®  {x  +  2){x  -  2){x  +  l){x  -  2)  =  0 
The  four  answers  to  this  equation  would  then  be 
X  =  2;  X  =  2;  X  =  —  2;  X  =^  —  1. 

EXERCISE    LXXIX. 

Solve  the  equations: 

1.  x^  +  2x^  =  9x  +  18.  6.  c?:^  -  27  =  19(a:  -  3). 

>  2.  a;3  —  12  =  4:c  —  3x\  7.  a;3  +  64  =r  5a;^  +  22rz:  +  8. 

3.  100  -  2)3  rir  42;2  -  25x.  8.  2^4 -13^2+36  =  3.^(0;'^- 9). 

4.  16a;  +  80  -  cr^  =  6x^  9.  ^4-13o;2-48  =  4o;3-64o;. 

5.  180  +  x^  =  6x^  +  36a;.  lo.  x^  -  2bx^  =  9x^^-  225. 


I 


FACTORABLE  EQUATIONS.  129 

144.  In  some  examples  (for  instauce,  in  the  tenth  of  the 
preceding  exercise)  easier  methods  of  factoring  may  be  seen 
on  simplifying  the  equation;  in  others,  the  simplified  form 
obtained  by  uniting  similar  terms  is  harder  to  handle. 


EXERCISE  LXXX. 

Solve  the  foUon^ing  equations : 

^'  +  ^^       o      ^    c,  1,1  1 

X  —  2  X       X  +  3       5 

x^+x-2     x^-x+2_l  2       _     3 

^'  3  I        ~6*  ^'  ^^^^~^'^' 

x^  -  6x  _2  x^  -  6  _dx-l 

^'  2x^  -1~3'  ^'        2      ~       4      • 

1  +  ^  +  2^:'^                                1        13 
4.  — -z =  2  +  3a:.       9.    -  = 7  +  x. 

1  ^  X  XX 

3a;  —  1_2:?;  —  1  x  _      3 

^'    x-2  ~  X  -3'  ^^'  3  ""  X  -3' 

11.  In  the  equation  x^  -\-  xy  -{-  2y^  =  74,  if  3/  =  5  what 
is  the  value  of  a;  ? 

1        1        14 

12.  In  the  equation  — I —  =-— ,  if  x  —  ^  what  is  the 

a;    '  ^       45 

value  oi  y  1 

13.  In  the  equation  4z{x  -\-  y)  =  3xy,  \i  x  =  4z  what  is 
the  value  of  y  ? 

3x 

14.  In  the  equation  x  +  2y  -\ =16  substitute  5  for 

X  and  then  find  the  value  of  y. 

15.  In  the  equation  xy  =  2400  substitute  100  —  x  for  y 
and  solve  for  x, 

16.  In  the  equation  -  -j-  -  =  2  substitute  2  —  x  tor  y 

x       y 

and  solve  for  x. 


130  ALomuA. 

17.  In  the  equation  35  =  3/i  -| ^^ — - — -d  substitute  2 

for  d  and  solve  for  n, 

18.  In    the    equation    s  =  an  -{-  d  ^ substitute 

s  =  119,  a  =  2,  d  =  6,  and  solve  for  7i, 

71 

19.  In  the  equation  5  =  —  (a  +  /)  substitute  5  =  105 , 
a=—  b^l=—  h-\-{n—  l)d,  d  =  ^,  and  solve  for  n 

,  In  the  equation  H=  - 
=  X  —  2,  and  solve  for  x. 


2Pp 
20.  In  the  equation  H  =  let  ZT  —  12,  F  =  x  -\-d, 


CHAPTER  VI. 

THE  FIRST  METHOD  OF  ELIMINATION. 

145.  The  equations  of  condition  *  already  studied  have 
been  true  only  on  condition  that  the  unknown  quantity 
had  a  particular  value,  or  one  of  two  or  three — one  of  a 
few — particular  values.     Thus  the  equation 

X  -\-7        =  2a;  —  3  is  true  only  when  x  =  10. 
x^  -|-  12     —7x         is  true  only  when  a;  =  3  or  4. 
{x^  —  Qy^  =  x^  is  true  only  when  ^'  =  3  or  —  3 

or  2  or  —  2. 

But  when  an  equation  contains  more  than  one  unknown 
quantity,  the  condition  implied  by  it  becomes  much  more 
indefinite.     The  equation 

2x  —  dy  =  6 

is  true  for  any  value  of  x  whatever;  but  only  on  condition 
that  when  x  has  any  value,  y  has  a  particular  value  which 
is  said  to  correspond  to  the  stated  value  of  x.  Suppose, 
for  instance,  x  =  6.  Then  2:^;  =  10  and  the  equation 
2a;  -  3^  =  6  becomes  10  -  3y  =  6.     Then 

10  =  6  +  3y  ;    4  =  32/  ;    H  =  y. 

If  x  =  5,  then  the  equation  is  true  only  on  condition 
that  y  =  1^, 

It  X  =  6,  y  must  equal  2. 

Ux=7,  y  =21;  iix  =8,  y  =  ^l;  it  x  =  9,  y  =  4:. 
*  See  p.  88. 

131 


13^  ALGEBRA. 

146.  There  is  no  end  to  the  solution  of  this  equation; 
we  might  continue  to  write  corresponding  values  of  x  and 
y  as  long  as  we  chose,  and  the  solution  would  still  be 
incomplete. 


The  Graphical  Method, 

147.  A  method  of  completely  exhibiting  the  solutions  of 
oh  is  kind  of  equations  will  now  be  described.  The  same 
method  is  often  used  for  other  purposes  in  science  and  in 
practical  affairs. 

148.  Different  values  of  one  letter,  like  x,  may  be  repre- 
sented by  equally  distant  points  along  a  straight  line; 
taking  a  particular  point  to  represent  zero,  positive  whole 

*  *     *  *   -     * »       * 

— 1 — I — I — ! — I — I — I — I — I — I — I — \ — I — \ — i — r — I — I — 1 — I — I — rn — 
.n_10-9-8' -7- 6-5-4-3-2-1    0   1    2    3   4    5    6   7    8    9  10 11 

Ftg.  1. 

numbers  are  represented  by  equally  distant  points  running 
out  towards  the  right,  while  negative  whole  numbers  are 
represented  by  equally  distant  points  running  out  towards 
the  left. 

Fractional  numbers  of  all  kinds  are  represented  by 
points  lying  between  the  whole  numbers;  thus  5  is  five 
spaces  to  the  right  of  zero,  ~  6  is  six  spaces  to  the  left  of 
zero,  10^  is  ten  and  one  third  spaces  to  the  right  of  zero, 
—  4^  is  four  and  one  half  spaces  to  the  left  of  zero,  etc. 

On  the  accompanying  diagram  the  stars  mark  approx- 
imately the  points  which  represent  —  10,  —  6,  —  4^,  0,  5, 
8^,  10^,  which  may  be  different  values  of  the  letter  x. 

The  only  numbers  that  have  no  points  on  this  scale  to 
represent  them  are  the  imaginary  numbers.* 

*  See  §  134. 


THE  FIRST  METHOD   OF  ELIMINATION,        133 

149.  Now  we  want  a  method  of  representing  at  the 
same  time  by  one  point  a  value  of  x  and  also  a  value  of  y. 
This  may  be  managed  by  taking  two  scale-lines  like  the  one 
just  described,  and  placing  them  perpendicular  to  each 
other  with  the  zero-points  together. 


X-- 




-7- 

. 

G 

P 

A 

0 

"  4: 

o 

^ 

i> 

•^ 

^ 

^ 

-i 

> 

1 

^a 

ns 

0/ 

X  I 

1 

t  _ 

3 

-^ 

'{ 

^-i 

I  - 

10 

1 

2 

3 

4 

6 

i 

-1 

-2 

o 

A 

-4 

pr 

-0 

-C 

-=-7 

__ 

_ 

_ 

_ 

_ 

--X 


Fig.  2. 

Call  the  horizontal  one  the  ^-scale,  or  axis  of  x^  and  the 
vertical  one  the  y-scale,  or  axis  of  y. 

Then  imagine  vertical  lines  drawn  through  each  point  of 
the  axis  of  x,  and  horizontal  lines  through  each  point  of 
the  axis  of  y, 

A  point  which  represents  x=  —  2 ;  y  =  4  must  be  the 
point  which  lies  opposite  —  2  in  the  axis  of  x,  and  opposite 
4  in  the  axis  of  y.  That  would  be  the  point  marked  P  in 
Fig.  2. 


134 


ALOBBRA. 


150.  The  points  in  Fig.  3  that  represent  the  following 
values  of  x  and  y  are  marked  with  the  corresponding 
Koman  numerals: 


lx  =  4:]y  =  2 
llx=.  —2]y  =  'd 
llVx  =  3 ;  «/  =  5 
IV^=z  6;^  =  -  3 

■a;  =  21; 


y- 


VII 


U=-3 
r^=  -  5; 
U=-3f 


-VH 


II.. 


4  -S  -2  -:i  0 


VI 


-4- 


'5 


III 


I-V- 


-X 


Fig.  3. 


151.   Eeturning  now  to  our  equation  2x  —  ^  —  6,  we 


: 

5 

i 

4 
o 

o 

1 

-I 

t- 

3-^ 

2-1 

I    i 

5    ^ 

( 

» 

r  6 

^  1 

) 

-5 

0 

j 

5 

10 

-1 

-2 

-3 







-4 

^5 



, 



L 



-X 


Fig.  4.— Partial  list  of  answers  to  the  equation  2x  —  Sy  =  6. 

find   the   following  set  of   answers,    represented  by  their 
respective  points,  running  up  from  left  to  right  on  Fig.  4; 


THE  FIRST  METHOD   OF  ELIMINATION,        135 


X  -  -  4.;     «/  =  - 

4| 

x  =  4;  y  =    1 

a;  =  -  3;     y  =  - 

4 

a;  =  5;  «/  =  1^ 

x=  -H;  y  ^  - 

3 

a;=  6;  y  =  3 

X  =  0;  y  =  -  2 

a;  =  7;  y  =  2| 

x=l;  y=  -1^ 

^  =  8;  y  =  3i 

x  =  2;  y  =^  ~    f 

a;  =  9;  «/  =  4 

X  =  3;  y  =  0 

An  Infinite  List  of  Answers. 

152.  It  would  be  found  on  trial  that  all  the  other  an- 
swers to  this  equation  would  lie  on  the  same  line,  i.e.,  their 
corresponding  points  would  fall  into  line  with  those  already 
put  down.     In  fact,  the  straight  line  drawn  through  two 


^ 

— 

— - 

—^ 

— 

— 

— 

— 

■~~ 

y 

^ 

/ 

/^ 

y 

/ 

y 

y 

/^ 

y 

y 

y 

^ 

y 

y 

y 

y' 

y 

y 

y 

y 

y 

^ 

y 

^ 

^ 

Fig.  5.— Complete  list  of  answers  to  the  equation  2a;  —  Sy  =  6. 

points  that  represent  any  two  answers  to  the  equation  is 
a  complete  list  of  all  possible  answers  to  the  equation.  In 
order  to  be  complete,  of  course,  not  only  the  two  axes,  but 
also  the  line  AB,  are  supposed  to  be  endless  in  extent. 


136 


ALGEBRA. 


Model  A. — In  a  similar  way  construct  a  complete  list  of 
answers  to  the  following  three  equations  (see  Fig.  6) : 
I    7a;  +    y  =14: 

II  2:^;  +  6^  =    3 

III  2ic  +  5y  =  10 


y 

1  r 

\ 

1 

lU 

\ 

I 

TT 

ei 

\ 

^ 

0 

\ 

"-^ 

\ 

\ 

I 

I"" 

-^ 

^ 

^ 

I 

■^ 

-^ 

^-^ 

-J 

^ 

"V 

^ 

X 

X 

_  p 

> 

0 

^ 

t 

^-v. 

^ 

I 

^ 

"^ 

\ 

\ 

\ 

-5 

\ 

T 

\ 

1  f\ 

\ 

-10 

■•' 

r 

.__. 

... 

Fig.  6. 


[Only  two  points  need  be  found;   then  a  straight  line 
can  be  drawn  through  them.] 


THE  FIRST  METHOD   OF  ELIMINATION. 


137 


EXERCISE  LXXXI. 

For  each  of  the  following  equations  construct  a  complete 
list  of  answers^  and  put  each  pair  on  a  separate  sheet :  * 

1.  x  +  2y  =^  ^]  dx  +  ly  =  17. 

2.  15a:  +  3y  =  39;  llx  —  y  =  31. 

3.  *Kx  -\-  4:y  —  41 ;  3a;  —  ^  =  4. 

4.  'Zx  -\-  y  =  11]  ^x  +  ly  =  22. 

6.  5a;  +  6^  =  51;  6a;  -  by  =  -  12. 

6.  2a;  -  «/  =  —  4;  7a;  +  6?/  =  62. 

7.  8a;  —  ^  =  27;  a;  +  8«/  =  44. 

8.  15a;  +  7^  =  162;  9a;  +  2y  =  84. 

9.  14a;  -  3^  =  44;  6a;  +  17^  =  92. 
10.  5a;  -  7^  ==  0;  Ix  +  by  =  74. 

153.  It  will  be  found  that  each  of  these  pairs  of  lines 
intersect  in  the  points  given  below;  the  points  are  num- 
bered to  correspond  with  the  pairs  of  equations: 


1. 

a;  =  1; 

y  =  2. 

6. 

a;  =  2; 

y  =  8 

2. 

a;  =  2; 

y-3. 

7. 

a;  =  4; 

y  =  5 

3. 

a;  =  3; 

y  =  5. 

8. 

a;  =  8; 

y  =  6 

4. 

a;  =  5; 

y  =  l. 

9. 

X  --=  4; 

y  =  4: 

5. 

a;  =  3; 

y  =  6. 

10. 

a;  =.  7; 

y  =  6 

154.  There  is  only  one  point,  then,  that  will  satisfy  two 
equations  of  condition  of  the  kind  used  so  far  in  this  chap- 
ter. That  is,  two  conditions  serve  to  determine  the  values 
of  two  unknown  quantities. 

155.  It  must  not  be  rashly  assumed  that  every  equa- 
tion with  two  unknown  letters  will  have  for  its  complete 

*  Paper  ruled  in  squares,  called  cross-section  paper,  can  be  bought 
of  dealers  in  drawing  materials. 


138 


ALGEBRA. 


list   of  answers   a    straight  line  ;    for  example,  Fig.    7 
gives  the  list  of  answers  for  (I)  x^  —  dy  -\-  10^  and  for 


y 

I 

\ 

\ 

y 

^ 

^^ 

v,_ 

L 

~y 

\ 

/ 

\ 

( 

\ 

/ 

\ 

\ 

-\ 

\ 

/ 

1 

— / 

\ 

v 

\ 

/ 

— ^ 

A 

\ 

^^ 

k 

/ 

^ 

^ 

^<: 

z^ 



Fig.  7. 

(II)  9:^2  4-  25?/2  =  225.  There  are  evidently  four  dif- 
ferent answers  that  would  make  both  these  equations 
true. 

To  construct  these  curves,  successive  values  of  y  are  sub- 
stituted in  each  equation,  and  the  values  of  x  calculated  by 
solving  the  equations  so  obtained.  Thus,  in  (I),  if  y  =  2, 
x  —  ^OYX  —  —^\  if^  =  —  2,  :^=2ora;  =  —  2;  ify=:l, 
X—  ±  i^Ts  =  3.606  or—  3.606;  and  so  on.  But  if  y  is 
taken  equal  to  — 4,  x  —  \^  —  "l  —  1.414i  or  —  1.414^, 
neither  of  which  can  be  represented  by  a  point  on  the  dia- 
gram. In  (II),  if  x  is  taken  greater  than  5,  or  y  greater 
than  3,  the  results  are  also  imaginary. 

Such  curves  as  these  form  the  subject-matter  of  an  ex- 
tensive and  very  interesting  branch  of  mathematics,  called 
Analytic  Geometry.  Not  much  time  can  be  profitably 
spent  on  them  here. 


THE  FIB8T  METHOD   OF  ELIMINATION.        139 

SIMULTANEOUS  EQUATIONS. 

156.  Model  B. — Six  horses  and  11  cows  sell  for  $810; 
13  horses  and  5  cows  sell  at  the  same  rate  for  $1190. 
Price  for  each  animal  ? 

Let  X  =  the  number  of  dollars  for  one  horse; 
y  =  the  number  of  dollars  for  one  cow. 
(i)  6x  +  lly  =  810 

(II)  Idx  +  5y  =  1190 

Model  C. — Six  pounds  of  tea  and  11  pounds  of  coffee 
sell  for  $9.23;  13  pounds  of  tea  and  5  pounds  of  coffee  sell 
at  the  same  rate  for  $11.90.     Price  of  each  ? 

Let  X  =  the  number  of  cents  for  one  pound  tea; 
y  =  the  number  of  cents  for  one  pound  coffee. 

(III)  6x  +  Uy  =  923 
(iv)  Ux  +  6y  =  1190 

In  the  four  equations  that  come  from  these  two  prob- 
lems X  and  y  do  not  have  the  same  meaning;  in  (i)  and 
(ii)  X  and  y  stand  for  a  number  of  dollars,  in  (iii)  and 
(iv)  for  a  number  of  cents.  When  the  unknown  letters 
in  two  equations  stand  for  entirely  different  quantities,  as 
in  (ii)  and  (iii),  they  must  be  treated  as  if  written  with 
different  letters;  the  terms  which  seem  to  be  similar  are 
really  n^ot  similar.  In  (i)  and  (ii)  we  should  find  that 
X  =  $80  and  y  =  $30 ;  while  in  (iii)  and  (iv)  we  should 
find  that  x  =  76  cents  and  y  =  4:3  cents. 

157.  When  in  two  equations  the  two  letters  stand  for 
the  same  unknown  quantities,  the  equations  are  called 
simultaneous.  Terms  having  the  same  letters  are  similar 
terms.  The  members  of  either  equation  can  be  added  to 
or  subtracted  from  the  corresponding  members  of  the  other 
equation,*  and  if  the  x  terms  (or  the  y  terms)  are  the  same, 

*  See  §  17. 


140  ALGEBRA. 

a  resulting  equation  may  be  obtained  free  from  x  (or  y). 
Thus,  for  Model  B  : 

(i)     6x+   11^  =  810 
(ii)    13:?;  +   5y  =  1190 

@    30x  +  bby  =  4050         (i)    X  5 
0  143^;  +  bby  =  13090     (ii)    X  11 
The  y  terms  are  now  alike.     Subtracting  (3)  from  0, — 
®  lUx  =  9040         ©  -  (D 
®  .T=:  80  ®    -^  113 

To  find  the  value  of  y  we  may  return  to  the  original 
equations,  (i)  and  (ii),  make  the  x  terms  alike  by  multiply- 
ing (i)  by  13  and  (ii)  by  6,  and  then  subtract  the  resulting 
equations.  Or,  since  we  know  now  that  x  —  80,  Qx  =  480, 
we  may  write,  instead  of  (i), 

®  480  +  lly  =  810      ®  substituted  in  (i) 
®  lly  =  330  ®  -  480 

(9)  y  =  30  ®  ^  11 

The  answer  to  Model  B,  then,  is  $80  for  each  horse,  and 
$30  for  each  cow.  In  the  same  way  we  find,  for  Model  C, 
75  cents  a  pound  for  tea,  and  43  cents  a  pound  for  cotfee. 

158.  This  is  the  method  by  which  the  points  of  crossing 
were  found  for  the  ten  pairs  of  lines  that  the  pupil  was 
asked  to  construct.  One  equation,  in  any  pair,  imposes 
upon  X  and  y  the  condition  that  their  point  must  lie  in  a 
certain  line;  the  other  equation  imposes  also  the  condition 
that  the  point  must  lie  in  another  line;  and  since  the  point 
must  lie  in  both  lines,  it  must  be  the  point  where  the  lines 
cross. 

169.  That  is,  taking  the  first  pair  of  equations  in  the  set 
of  ten  mentioned,  the  two  conditions  contained  in  the  equa- 
tions x-{-2y  =  5;  3x  -{-Hy  =  17  are  the  same  as  these  two 
conditions:  ^  =  1 ;  y  =  2,  The  algebraic  work  consists  in 
reducing  the  given  conditions  to  the  form  of  particular 
values  for  x  and  y. 


THE  FIRST  METHOD   OF  ELIMINATION,        141 

Elimination  by  Combination. 

160.  The  process  of  getting  rid  of  an  unknown  letter  is 
called  '^eliminating^'  that  letter.  The  method  used  above 
is  called  Elimination  by  Combination.  Other  methods  will 
be  described  later. 

EXERCISE   LXXXII. 

Solve  the follo2vi7ig  equations:^ 

1.  2:c  +  7^  =  59 ;  3a;  +  %  =:  43. 

2.  bx  +  ^y  =  34;  ^x  +  ^  =  24. 

3.  13^  +  17^=107;  '^x-\-y=10, 

4.  14a;  +  9t/  =  100;  7a;  +  2^  =  30. 
6.  a;+15^=37;  3a;  +  7^  =  35. 

6.  15a;  +  19?/  =  79;  35a;  +  lly  =  157. 

7.  6a;  +  4?/  =  90;  3a;  +  Iby  ^  123. 

8.  39a:  +  21  y  =  213;  52a;  +  29y  =  249. 

9.  72a;  +  Uy  =  170;  63a;  +  7y  =  112. 
10.  101a;  +  26y  =  886 ;  103a;  +  39y  =  941. 

tu.  2a;  +  7^  =  49;  6x  ~  6y  =  17. 

12.  7a;  -  4y  :=:=  2;  25a:  -  13^  =  24. 

13.  X  -{-y  =  90;  x  —  y  =  13. 

14.  4:X+9y  =  60;  7x  -  lly  =  22. 

15.  bx  —  3y  =  17;  V2y  —  lx=  23. 

16.  171a;  -  213^  =  642;  114a;  -  326?/  =  244. 

17.  9^  -  7a;  =  43;  15a;  -ly  =  43. 

18.  12a;  +  7y  =  176;  3y  -  19a;  =  3. 

19.  43a;  +  2y  =  266;  12a;  -  17^  =  4. 

20.  5a;  +  9y       188;  13a;  -  2y  =  57. 

21.  4a;  +  3^  =  22;  3a;  +  5^  ==  11. 

22.  bx  -  2y  =  bl;  llx  -  3y  =  162. 

*  That  is,  reduce  the  given  conditions  to  the  simplest  form. 

f  It  may  happen  that  the  x  terms  (or  the  y  terms)  are  of  opposite 
signs  in  the  two  equations ;  in  that  case  the  equations  must  be  added 
to  get  rid  of  that  letter. 


14:2 


ALGEBRA. 


31. 


32. 


33. 


34. 


36. 


23.  ^x  -  by  =^  51;  bx  ~\-ly  =  39. 

24.  '^y  —  6x  =  143;  3:^;  +  5y  =:  89. 

25.  4:?;  +  lly  =  144;  82:  +  17y  =  198. 

26.  2a;  —  7^  =  8;  3y  —  9:^:  =  21. 

27.  17^;  +  l^y  =  59;  19^;  -  3^  =  148. 

28.  8a;  +  3^  =  3;  4:^  +  3?/  =  1. 

29.  59y  -  17a;  ==  123;  2a;  -  13y  =  -  17. 

30.  3a;  +  2y  =  42;  13a;  +  23y  =  225. 


80x  +  50y  =  22190; 
70a;  =11(60^). 


C2x      3y 
3  "^  4 

=  700; 

J/ +100 

_  bx 
~   6" 

ra;  +  2/  = 

7; 

I5  +  X 

3 

^9  +  y 

4* 

x-\-y  = 

15 
16* 

x-y  = 

13 
16' 

C2x      y 
3  "^2 

=  60; 

=  8. 

36. 


37. 


38. 


40. 


^  +  60  =  ^; 

x-\-y  =:  120. 
a;  +  2_  1 

X      _1 
"^3- 


+  3 

a;+3_ 
2/+3~ 


a;  —  3 


39.  a;  —  y  =  5 ; 


12  -  a;  _  2 

17^"  7* 


|a;  +  y  ==  7; 

|l0a;  +  y  +  9  =  10?/- 


EXERCISE    LXXXIII. 

1.  For  8  cows  I  get  $20  more  than  I  pay  for  40  sheep; 
and  for  16  sheep  I  pay  $21  more  than  I  get  for  3  cows. 
Price  of  each  ? 

2.  Find  two  numbers  whose  difference  is  -^^  of  their 
sum,  and  3  less  than  \  of  the  larger  number. 

3.  In  10  hours  A  walks  1  mile  less  than  B  does  in  8 


THE  FIRST  METHOD  OF  ELIMINATION,        143 

hours;  and  in  6  hours  B  walks  1  mile  less  than  A  does  in 
8  hours.     How  many  miles  does  each  walk  per  hour  ? 

4.  If  A^s  money  were  increased  by  36  cents  he  would 
have  3  times  as  much  as  B;  if  B's  money  were  increased 
by  5  cents  he  would  have  half  as  much  as  A.  How  much 
has  each  ? 

5.  A  pound  of  tea  and  6  pounds  of  sugar  cost  72  cents; 
if  sugar  were  to  rise  50  per  cent  and  tea  10  per  cent,  the 
same  quantity  would  cost  84  cents.  Price  of  tea  and  of 
sugar  ? 

6.  Find  two  numbers  such  that  3  times  the  greater 
exceeds  twice  the  less  by  29,  and  twice  the  greater  exceeds 
3  times  the  less  by  1. 

7.  Three  men  and  16  boys  earn  $107.25  in  h\  days, 
and  4  men  with  10  boys  earn  $192.50  in  11  days.  What 
are  the  daily  wages  of  men  and  of  boys  ? 

8.  A  farmer  bought  land  for  $7500,  part  at  $80  per  acre 
and  part  at  $50  per  acre.  The  cheaper  he  sold  at  a  loss  of 
10  per  cent,  and  the  dearer  at  a  gain  of  10  per  cent.  On 
the  whole  he  profited  $50  by  the  transaction.  How  much 
land  did  he  buy  ? 

9.  If  I  divide  the  smaller  of  two  numbers  by  the  greater, 
the  quotient  is  .36  and  the  remainder  .64;  if  the  greater  is 
divided  by  the  smaller,  the  quotient  is  2  and  the  remainder 
27.     Find  the  numbers. 

10.  The  sum  of  the  ages  of  a  father  and  son  will  be 
doubled  in  25  years;  the  difference  of  their  ages  is  \  of 
what  the  sum  will  be  in  20  years.     How  old  are  they  ? 

11.  Find  two  numbers  such  that  3  times  the  greater  ex- 
ceeds \  the  less  by  439,  and  3  times  the  less  exceeds  \  the 
greater  by  49. 

12.  Eight  years  ago  A  was  5  times  as  old  as  B;  7 
years  hence  he  will  be  only  twice  as  old.  How  old  are 
they  now  ? 

13.  A  merchant  offers  me  8  pounds  of  black  tea  and  24 


144  ALOBBIRA. 

pounds  of  green  tea.  for  110,  or  14  pounds  of  black  tea  and 
17  pounds  of  green  tea  for  the  same  sum.  If  it  makes  no 
difference  to  him  which  offer  I  accept,  what  are  his  prices 
per  pound  ? 

14.  A  and  B  buy  a  horse  for  $550;  A  can  pay  for  it  if 
B  will  advance  J  the  money  he  has  in  his  pocket;  and  B 
can  pay  for  it  if  A  will  advance  J  the  money  in  his 
pocket.     How  much  has  each  ? 

15.  A  sum  of  money  was  divided  equally  among  a  cer- 
tain number  of  persons;  if  there  had  been  10  more  persons, 
each  would  have  received  $2  less;  if  there  had  been  10  less, 
each  would  have  received  $3  more.  How  many  persons, 
and  the  share  of  each  ? 

16.  If  a  certain  lot  of  land  were  8  feet  wider  and  2  feet 
longer,  it  would  contain  960  square  feet  more;  if  it  were 

2  feet  narrower  and  8  feet  shorter,  it  would  contain  760 
square  feet  less.     What  is  its  area  ? 

17.  A  farmer  bought  eggs  at  2  for  5  cents,  and  others  at 

3  for  8  cents;  wishing  to  sell  them  to  a  relative  at  cost,  he 
named  a  price  of  31  cents  per  dozen,  but  found  he  would 
lose  5  cents;  his  relative  then  suggested  5  for  13  cents, 
and  the  farmer  accepted  that,  but  made  a  profit  of  10  cents 
on  the  transaction.  How  many  dozen  were  there  of  each 
kind? 

18.  It  takes  me  16  times  as  long  to  walk  around  the 
edge  of  a  long  rectangular  field  as  to  walk  directly  across 
it;  and  the  next  field,  which  is  30  feet  wider  and  150  feet 
shorter,  has  the  same  area.  What  are  the  dimensions  of 
each  field  ? 

19.  A  cruiser  386.8  feet  long  passes  a  battleship  going 
in  the  same  direction  in  1.12  minutes;  returning,  she 
passes  the  same  battleship  in  12  seconds.  The  length  of 
the  battleship  is  352.4  feet.  Supposing  each  ship  main- 
tains the  same  speed  throughout,  how  many  miles  per  hour 
do  they  go  (5280  feet  to  the  mile)  ? 


TEE  FIB8T  METHOD   OF  ELlMmATlON.        146 


21. 


22. 


I 


20.  Two  men  who  lived  in  towns  70  miles  apart  started 
at  the  same  time  and  rode  towards  each  other;  when  they 
met,  they  exchanged  horses  and  continued  as  before.  One 
horse  could  go  10  miles  per  hour,  the  other  only  8.  The 
slower  horse  arrived  while  the  faster  horse  was  still  5  miles 
from  the  journey's  end.  Find  the  time  each  traveller  took 
on  the  road. 

Find  the  fractions  descriled  as  follows  : 

If  4  be  added  to  the  numerator,  the  value  of  the 

fraction  will  become  1; 
If  3  be  added  to  the  denominator,  the  value  of  the 

fraction  will  become  i. 
If  1  be  subtracted  from  the  numerator,  the  value 

of  the  fraction  will  become  -|; 
If  11  be  added  to  the  denominator,  the  value  of 

the  fraction  will  become  ^. 
If  8  be  added  to  the  numerator,  the  value  of  the 

fraction  will  become  f ; 
If  8  be  added  to  the  denominator,  the  value  of  the 

fraction  will  become  i. 
If  17  be  added  to  numerator  and  to  denominator, 

the  value  of  the  fraction  will  become  |; 
If  1  be  subtracted  from  numerator  and  from  de- 
nominator, the  value  of  the  fraction  will  become  i. 
If  3  be  taken  from  the  numerator  and  1  be  added  to 
the  denominator,  the  value  of  the  fraction  will 
become  J; 
If  3  be  added  to  the  numerator  and  5  to  the  denomi- 
nator, the  value  of  the  fraction  will  become  -J. 
Find  two  fractions  with  numerators  4  and  11  re- 
spectively, such  that  their  sum  is  2i|,  and  if  their  denomi- 
nators are  interchanged  their  sum  is  2||. 

27.  A  fraction  is  such  that  when  it  is  multiplied  by  f 
the  sum  of  its  terms  is  107;  if  8  is  added  to  each  of  its 
terms  the  value  of  the  fraction  becomes  |. 


23.  ^ 


24. 


25. 


26. 


146  ALGEBRA. 

28.  A  proper  fraction  is  such  that  if  the  numerator  is 
halved  and  the  denominator  increased  by  3  its  value  be- 
comes \]  if  the  fraction  is  multiplied  by  |  the  difference  of 
its  terms  will  be  53. 

29.  The  sum  of  the  terms  of  a  fraction,  divided  by  their 
difference,  gives  2  for  a  quotient  and  5  for  a  remainder. 
If  the  fraction  were  divided  by  2  the  sum  of  its  terms 
would  be  53. 

30.  Two  fractions  have  denominators  20  and  10  respec- 
tively. The  fraction  formed  from  these  two  by  taking  for 
its  respective  terms  the  sums  of  the  corresponding  terms  of 
these  two  fractions  is  f ;  and  the  fraction  similarly  formed 
by  taking  the  differences  is  -f. 

Find  the  numbers  described  as  follows : 

31.  A  number  divided  by  the  sum  of  its  digits  gives  7f 
for  a  quotient;  if  54  be  subtracted  from  the  number,  the 
digits  are  reversed. 

32.  If  I  divide  a  number  by  the  sum  of  its  digits,  the 
quotient  is  6  and  the  remainder  7;  but  if  I  invert  the 
order  of  the  digits  and  then  divide  by  the  sum  of  the  digits, 
the  quotient  is  4  and  the  remainder  6. 

33.  A  number  exceeds  5  times  the  sum  of  its  digits  by  8; 
if  the  order  of  the  digits  were  reversed,  the  number  would 
be  15  less  than  7  times  the  sum  of  the  digits. 

34.  A  number  is  4  times  the  sum  of  its  digits;  and  if  27 
be  added  to  the  number,  the  order  of  the  digits  is  reversed. 

35.  A  number  is  2  more  than  5  times  the  difference  of 
its  digits;  if  its  digits  are  reversed,  the  resulting  number 
is  8  times  the  sum  of  its  digits. 

36.  Two  numbers  have  the  same  digits  in  opposite  order; 
the  difference  of  the  numbers  is  3  times  the  sum  of  the 
digits;  and  the  sum  of  the  digits  is  2  more  than  2|  times 
the  difference  of  the  digits. 

37.  If  45  be  subtracted  from  a  number,  the  digits  are 


THE  FIRST  METHOD  OF  ELIMINATION,        147 

reversed;  but  the  same  result  might  have  been  obtained  by 
subtracting  7  from  tlie  number  and  then  dividing  by  3. 

38.  Two  numbers  which  have  the  same  digits  in  opposite 
order  differ  by  18,  and  the  smaller  number  is  4  times  the 
sum  of  the  digits. 

39.  Subtracting  27  from  a  number  reverses  the  digits; 
in  the  scale  of  8,  instead  of  the  decimal  scale,  the  number 
would  be  18  less. 

40.  If  a  certain  number  is  divided  by  the  sum  of  its 
digits,  the  quotient  is  4  and  the  remainder  6 ;  if  the  digits 
are  reversed,  the  resulting  number  divided  by  5  gives  4 
more  than  the  sum  of  the  digits. 

41.  A  number  consists  of  three  digits,  the  middle  digit 
being  zero.  If  the  digits  are  reversed,  the  number  is 
increased  by  396 ;  if  only  the  second  and  third  digits  change 
places,  the  number  is  increased  by  3  more  than  6  times  the 
sum  of  its  digits. 

42.  A  number  of  three  digits,  the  middle  digit  being  6, 
has  its  digits  reversed  if  one  adds  to  it  22  times  the  sum  of 
its  digits ;  and  the  number  so  formed  is  72  less  than  double 
the  original  number. 

43.  A  number  has  four  digits,  of  which  the  second  is  6 
and  the  fourth  is  3 ;  reversing  the  digits  increases  the  num- 
ber by  909 ;  but  if  only  the  first  and  third  digits  change 
places,  the  number  is  increased  by  2970. 


148  ALQBBBA. 

Solving  for  Reciprocals. 

161.  In  equations  like  the  following  it  is  better  to  make 
the  X  terms  (or  the  y  terms)  alike  without  first  clearing 
of  fractions : 

Model  D. 


^32        3 

^5,4        11 

^6        4        4 

®  X  3 

®l  =  k 

(D-® 

0)  2l  =  x 

®  X  31a; 

similarly  for  y. 

EXERCISE    LXXXIV. 

1.  A  and  B  together  can  do  a  piece  of  work  in  8|  days; 
if  A  worked  3  days  and  B  5  days,  only  half  of  the  work 
would  be  done.  How  long  for  A  alone  ?  How  long  for 
B? 

2.  Five  boys  and  10  men  could  do  a  certain  job  in  3 
days;  one  man  and  one  boy  would  take  24  days  to  do  it. 
How  long  for  one  man  alone  to  do  the  work  ?  One  boy 
alone  ? 

3.  Twenty-four  pails  of  water  and  20  cans  of  milk  will 
just  fill  a  certain  tank;  6  pails  and  14  cans  will  half  fill 
it.     How  many  pails  to  fill  it  ?     How  many  cans  ? 

4.  Two  pipes  fill  a  cistern  in  20  minutes;  if  one  of  the 
pipes  were  twice  as  large  *  and  the  other  half  as  large,  the 
cistern  would  be  filled  in  15  minutes.  How  long  for  each 
pipe  ? 

*  The  word  **  large  "  here  refers  not  to  the  width  but  to  the  capac- 
ity of  the  pipe. 


THE  FIRST  METHOD   OF  ELIMINATION.        149 

5.  Two  pipes  can  fill  a  cistern  in  5  minutes;  if  one  of 
the  pipes  is  closed  half  the  time  it  will  take  7|^  minutes. 
How  long  for  each  pipe  alone  ? 

6.  A  and  B  could  dig  a  well  in  12  days;  but  at  the  end 
of  the  third  day  B  quits,  so  that  the  job  lasts  A  6  days 
longer.     How  long  for  each  alone  ? 

7.  After  working  2  days  on  a  certain  job  with  B,  A  says 
to  him,  '^I  can  finish  this  job  alone  in  10  days.^^  B  re- 
plies, *^If  we  work  together  one  more  day,  I  can  finish  it 
alone  in  5  days/^  If  what  they  say  is  true,  how  long  will 
it  take  each  alone  to  finish  the  job  ? 

8.  A  can  row  11  miles  down-stream  for  every  7  against 
the  stream;  he  rows  down-stream  for  3  hours,  then  rows 
back,  and  at  the  end  of  the  6  hours  he  is  5  miles  from 
his  starting-place.  How  fast  does  he  row,  and  how  fast 
does  the  stream  flow  ? 

9.  B  rows  9  miles  down-stream  in  45  minutes;  he  rows 
back,  near  the  bank,  where  the  current  is  only  half- 
strength,  in  one  hour  and  a  half.  Speed  of  boat  and  of 
stream  ? 

10.  C  rows  6  hours  down-stream  and  15  hours  up- 
stream, covering  in  all  72  miles.  His  speed  up-stream  is 
f  of  his  speed  in  still  water.     Speed  of  stream  ? 

11.  D  rows  down-stream  for  an  hour  and  a  half,  but  it 
takes  him  3  hours  to  row  back.  He  rows  in  the  first  3 
hours  12  miles.     Speed  of  boat  and  of  stream  ? 

12.  A  and  B  run  a  mile  race;  A  gives  B  12  seconds  start 
and  beats  him  by  44  yards;  then  A  gives  B  165  yards  start, 
and  is  beaten  by  10  seconds.     Speed  of  each  ? 

13.  A  traveller  started  on  a  journey  of  330  miles,  having 
4  hours  and  12  minutes  to  spare  on  a  connection  he  ex- 
pected to  make  at  the  end  of  that  journey ;  but  an  accident 
occurred  when  he  was  2  hours  out,  which  not  only  held  up 
the  train  for  2  hours,  but  diminished  its  speed  for  the  rest 
of  the  run.     Then  he  missed  his  connection  by  just  3  min- 


160  ALGEBRA. 

utes.  If  the  accident  had  happened  6  miles  further  on,  he 
would  have  been  barely  in  time.  How  fast  did  the  train 
go  before  and  after  the  accident  ? 

14.  A  was  sent  to  a  town  147  miles  away,  and  7  hours 
later  B  was  sent  after  him.  After  travelling  71  miles  B 
was  handed  a  letter  to  deliver  to  a  person  living  17  miles 
out  of  the  town  to  which  A  had  been  sent.  He  overtook 
A  just  as  he  was  entering  the  town,  and  handed  him  the 
letter;  the  letter  was  delivered  just  9  hours  and  40  minutes 
after  B  received  it.     Speed  of  A  and  of  B  ? 

Where   Elimination   Fails. 

162.  In  studying  equations  of  one  unknown  letter,  we 
found  that  the  given  equation  could  be  reduced  to  a  very 
simple  equation  giving  a  particular  value  for  that  letter. 
That  simple  equation  was  the  condition  that  the  given 
equation  should  be  true. 

We  afterwards  found  that  if  there  were  two  unknown 
letters  in  an  equation  of  condition,  another  equation  had 
to  be  given  before  we  could  get  particular  values  of  x 
and  y.  When  only  one  equation  is  given,  either  of  the 
two  letters  can  have  any  numerical  value;  there  must  be 
given  two  equations  of  condition  before  their  values  are 
limited  to  a  particular  set. 

163.  In  this  connection  the  following  problem  is  in- 
structive : 

Model  E. — A  certain  number  of  two  digits  is  equal  to  4 
times  the  sum  of  its  digits ;  if  the  digits  are  reversed,  it  is 
equal  to  21  times  the  difference  of  its  digits. 
Let  X  =  the  tens  digit; 
y  =  the  units  digit. 

([)  10x  +  y  =  4.{x  +  y) 

®lOy  +  x  =  21(^  ^  y) 

(D  lOo;  -^^  y  =^  4:X  -{-  4:y       same  as  Q 


THE  FIRST  METHOD   OF  ELIMINATION,        151 
®  %x  —  ^  —  0  (D   -  4^  ~  4y 

®  \0y  +  X  =  21x  —  21y    same  as  © 

®  31^  -  20^  =  0  ®  -  21x  +  21y 

If  we  compare  equations  ®  and  ®  we  shall  see  that 
both  cannot  at  the  same  time  be  true ;  ®  says  that  x  —  \y, 
while  ®  says  that  x  =  ^^y. 

164.  Such  equations  are  called  inconsistent,  and  of 
course  cannot  be  used  to  determine  the  values  of  the 
unknown  quantities.  They  point  to  some  mistake  in  the 
statement  of  the  problem,  or  in  the  pupil's  understanding 
of  it. 

165.  We  tacitly  took  for  granted,  in  forming  equation 
@,  that  the  tens  digit  was  larger  than  the  units.  Let  us 
see  if  that  is  the  source  of  the  inconsistency. 

®  10a;  -}-  y  =  4:X  -}-  4:y 
©  10^  +  ^  =  21y  -  21:r 

®  22a;  -  11^  =  0  @  +  2I2:  -  21y 

(T)  2x  -  y  =  0  ®-^ll 

]N"ow  the  equations  ®  and  @  are  consistent, — too  much 
so  in  fact ;  for  the  second  equation  reduces  to  one  precisely 
like  the  first.  In  this  case  also,  then,  we  cannot  determine, 
from  the  two  equations  given,  the  particular  values  of  x 
and  y. 

166.  In  all  equations  of  this  kind,  then,  we  must  require 
that  they  be  consistent,  that  is,  that  both  cak  be  true  at 
the  same  time;  and  that  they  be  independent,  that  is,  that 
both  must  not  reduce  to  the  same  equation. 

Non-Algebraic  Conditions. 

167.  It  happens  sometimes  that  a  problem  is  given,  like 
the  above,  which  implies  only  one  algebkaic  condition, 
but  still  the  answer  is  restricted  to  a  few  or  perhaps  even 


162  ALGEBRA. 

to  only  one  particular  value,  by  a  condition,  implied  in 
the  statement  of  the  problem,  such  that  it  cannot  be  stated 
algebraically. 

In  Model  E  our  digits  must  be  from  the  nine  Arabic 
numerals;  we  found  by  trial  that  our  equations  were  in- 
consistent and  the  problem  unsolvable  unless  the  second 
digit  were  the  larger;  and  our  algebraic  conditions  both 
reduced  to  the  fact  that  the  second  digit  was  twice  the 
first.     That  leaves,  for  the  only  possible  solutions, — 

X  =  Ij  ^  =1  2;  the  number  12 

^  r=  2;  y  =  4;  the  number  24 

cc  =  3;  y  =  6;  the  number  36 

a;  =  4;  y  =  8;  the  number  48 

Four  particular  answers  to  the  problem. 

168.  Equations  which  do  not  give  a  solution  from  the 
algebraic  conditions  alone  are  called  indeterminate  equa- 
tions; and  the  problems  which  give  rise  to  such  equations, 
although  they  may  contain  enough  non-algebraic  con- 
ditions to  determine  the  answer,  are  often  called  indeter- 
minate problems. 

More  Than  Two  Unknown  Letters. 

169.  The  equation  3x  -\-  4=7/  -{-  Qz  =  41  has  three  un- 
known letters  in  it.  If  we  take  the  value  of  x  to  be  3,  the 
equation  reduces  to  2y  +  3,^  =  16;  and  if  we  take  the 
equation  x  -{-  2y  -{-  dz  =  19  and  eliminate  x  thus: 

(£}  ^x  +  iy  +  6z  =  41 

(2)  X  +  2t/  +  dz    =19 

®  3x  +  61/  +  9z  =  57    (2)   X  ^ 

®  2y  +  ^z  =  U  ®  -  ® 

we  come  to  the  same  result. 

In  this  case  we  have  imposed  the  same  condition  in  two 
different  ways;    and  from  our  experience  with  equations 


THE  FIRST  METHOD  OF  ELIMINATION.        153 

like  2y  +  82;  =  16  we  know  that  the  values  of  y  and  z  can- 
not be  determined  unless  we  have  one  more  condition. 

Counting  the  given  equation  of  condition,  the  second 
that  we  imposed  at  random  (since  a;  =  3  and  x  -\-'ly  -\-Zz 
=  19  are  not  independent  conditions  we  have  only  im- 
posed one),  and  the  third  that  we  know  we  must  have, 
there  are  three  conditions  necessary  to  determine  three 
unknown  values. 

If  the  values  turn  out  to  be 

:?;  =  3;     y  =  5;     0  =  2, 

or  whatever  else  they  are,  these  answers  would  themselves 
be  THKEE  independent  equations;  and  to  get  them  we 
must  have  three  independent  equations  to  start  with. 

170.  Whatever  the  number  of  unknown  letters,  the  same 
number  of  independent  equations  of  condition  is  required 
to  determine  their  values. 

Elimination  with  Three  Letters- 

171.  Model  F. 

©  32;  +  %  +  5:2  =  39  ^ 

(D     2:  4-  2y  -f-  3;2  =  19  V  Three  given  conditions, 

(3)  4a;  +  5y  +  7^  =  51  j 

®  32;  +  6^  +  9^  =  57         ©  X  3 
(5)  2^  +  4^  =  18         ®  -  ® 

d)  ^  +  2^  =9         ®-f-2 

We  have  now  eliminated  x  from  the  first  two  equations. 
We  might  have  eliminated  y  or  ^  from  the  same  two  equa- 
tions, or  we  might  have  taken  ®  and  (3)  or  @  and  (3), 
instead  of  the  first  pair. 

Now  that  we  have  started  with  ^•,  however,  we  must  con- 
tinue to  eliminate  this  letter;  (6)  is  an  equation  with  y  and 
z  for  letters,  and  we  need  another  equation  to  determine 
their  values;  so  we  must  eliminate  x  again;  and  we  must 
take  a  different  pair  of  equations,  either  Q  and  (3)  or  (2) 


154  ALGEBRA, 

and  (3),  or  else  our  second  equation  in  y  and  z  will  not  be 
independent  of  (6). 

®  4a;  +  8^  +  12;^  =  76         ©  X  4 

(D  3^  +  5^  =  25         ®  ~  ® 

(6)  and  (D  are  two  independent  equations  with  two  unknown 
letters;  and  the  rest  of  the  solution  is  familiar  ground. 

®        3?/  +  6;^  =  27    ©  X  3 

@  ^  =  2     ®  -  © 

@  y  _[_  4  =  9     (g)  substituted  in  ® 

®  y  =  5     (0)  -  4 

@  2;  +  10  +  6  =  19    (10)  and  (iD  subst.  in  © 

(U)  a;  =::  3     @  -  16 

^^5.  a;  =  3;y  —  5;;2;  =  2. 
172.  When  we  have  obtained  two  equations  from  which 
one  of  the  unknown  quantities  is  eliminated,  these  are  called 
the  equations  of  the  new  set.     In  the  preceding  solution 
equations  ©  and  ©  are  the  equations  of  the  new  set. 

EXERCISE   LXXXV. 

i^x  +  ly-2z-l'i,  iy  -  x-\-  z  =  ~  6, 

1.  \sx  -\-  ^  -\-  z  =  11,        6.  \z-y-x=—^^, 
(  ^-  -  4^  +  10^  =  23.  ix  -\'  y  +  z  =  35. 
i^x~2y-\-bz  =  16.  i  3.T  -  2y  +  5^  =  26. 

2.  •<  32;  -  2y  +  4^  =  -  10.  7.  \  x  -  2y  +  3^  =  6. 

(  4:z;  +  2?/  -  5^  =  -  2.  (  2a;  +  3^/  -  4^;  =  20. 

r  5;^;  _j-  3y  _  6;^;  =:  4.  r  4:?;  -  3?/  +  2;^  r=  40. 

3.  X%x  ~  y  ~\-'%z  =  8.  8.  \hx-\-^-lz  =  47. 
(  ^  -  2^  +  20  =  2.  (  92;  +  8y  -  3;^  =  97. 
f  2a;  -  2?/  +  ;2  =  1.                 (  3a;  +  2y  +  ;^  =  23. 

4.  -j  2a;  +  3y  —  0  =  5.  9.    -j  5a;  +  2^  +  4;^  =  46. 

(  a;  4-  y  +  ;^  =  6.  (  10a;  +  5^  +  4^  =  75. 

(  2a;  +  3?/  +  42;  =  20.  pa;  -  6^  +  4^  =  15. 

5.  -j  3a;  +  4y  +  5;^  nr  26.  10.  K  7a;  +  4y  -  3^;  =  19. 
(  3a;  +  5y  +  60  =  31.  (  2a;  +  1/  +  6^  =  46. 


THE  FIRST  METHOD   OF  ELIMINATION.        155 


Solve  for  reciprocals  first :  * 


11. 


12. 


11       1 
^  +  ,-  +  7^ 

=  36. 

^2        2 

7 
"60* 

■< 

^  +  ^-1  =  28. 

X    '  y        z 

13. 

< 

3        4        2 

^  '^y~"z' 

7 
"30* 

i  +  3^  +  2l  =  ^«- 

5        3        2 

3 

~10* 

X  ^  y        z 

X    '  y    '    z 

=  36. 

, 

X      y    '   z 

14. 

- 

1   ,   ^^i 

=  2S. 

^-^  +  ^  =  23. 

11        1 
^x  +3^+2^ 

^  20, 

15.     -< 

'  3        4 

iz;  ~5^~^ 
1         1 

1  38 
~z~  5* 

2  83 
^~12' 

* 

4 
^5a; 

1 

"2^  + 

4       121 

;2  ~  20' 

172 

1.  Model  G.— I 

'rom  the  fou 

r 

equal  expressic 

>ns  in 

continued  equation 

^{x  -\-  z  -  b)  =  y  -  z  ^'^x  -11  =  ^  —  {x  +  2z) 
we  could  form  six   different  equations,  but  only  three  of 
them  would  be  independent ;  from  these  three  the  values  of 
x,  y,  and  z  can  be  determined. 

(^  :^(x-\-  z  -  b)  =  y  -  z 

@  y  —  z  =  2x  —  11 

®  ^x  -  11  =  ^  -  {x  +  2z) 


®x  +  z  —  b=^^  —  'Zz 
®  X  -  ^  -\-^z  =  b 
(§)  2x  —  y  +  z  =  11 
®  32;  +  2^  =  20 

*  See  §  161. 


®    X2 

®  -  21/  +  2^  +  5 
®  -y  +  Z+11 
@  +  :«;  -  2;2  +  11 


156  ALGEBRA, 

Here  are  three  equations,  ®,  (6),  and  (7),  for  ttiree 
unknown  letters;  but  one  of  the  equations  has  only  two 
unknown  letters  in  it.  The  letter  missing  from  that  is  y, 
so  we  combine  ®  and  (6)  to  get  rid  of  y,  and  thus  get  two 
equations  with  two  unknown  letters  [®  and  (9)]. 

®  4^  -  2y  +  22;  =  22    (6)   X  2 
®  3a;  -  ;2  =  17  ®  -  ® 

(7)  ^x  +  22;  =  20  ]  ^^ 

^^  ^  New  set 


]■ 


®  'dx  -  z  =  11 
Whence  we  obtain  x  =  Q\  y  =  2]  z  =  \,     Ans. 


EXERCISE    LXXXVI, 

1.  a;  +  20  =  ^  +  10  =r  2;^  +  5  =  110  -  (3^  +  ;^). 

y  +  z       z  -\-  X       X  4-  y 

2.  ^^  =  ~-  =  -^;     x  +  y  +  z  =  27, 

3.  ^--, —  =  ^-^ —  =  bz  —  ^x\     y  +  z  =  2x  +  1. 
^,  X  +  2y  =  6z  -  lOx  =  y  +  z  =z  600. 

5.  x-^-  =  Q;     y  _  ^  ^  8;     ^  -  |  .:.  10. 

2,1         33        2        ^1,1        4 

6.  -  +  -  =  5-; =2;      -  +  -  =  -. 

a;        ^        2z      z        y  x        z        3 


7.  2a;-3y=8;     ^~3;^=-ll;     x  -  2y -\- ^z  =  Yt. 

1,2^34  .34^ 

8.  -+-  =  5;     -— _zzr_6; =5. 

X       y  y       ^  z       X 

2x  -  y  _Zy  -^  2z  _x  ^  y  -  z  _ 


9. 
10. 


3  4  5  ~ 

X  ~  y  __y  —  ^ __ X  +  z _x  -\-  10 

3  4~  5  ^i2~"' 


THE  FIRST  METHOD   OF  ELIMINATION,        157 
EXERCISE    LXXXVII. 

1.  A  storekeeper  exchanged  with  a  neighbor  2  bushels 
of  oats  and  rye,  mixed  half  and  half,  for  If  bushels  of  corn ; 
and  he  told  the  neighbor  that  of  two  bills  that  he  sent  out 
for  corn,  oats,  and  rye,  the  first  was  for  5,  6,  and  8  bushels 
respectively,  and  amounted  to  $10.30;  the  second  was  for 
3,  5,  and  8  bushels  respectively,  and  amounted  to  $8.75. 
Price  of  each  kind  of  grain  ? 

2.  Suppose  you  and  I  together  had  55  cents;  you  and 
some  friend  of  ours  had  62  cents;  while  the  three  of  us 
had  97  cents.     How  much  would  each  of  us  have  ? 

3.  A  pays  to  B  and  C  as  much  as  each  of  them  has;  then 
B  pays  to  A  and  C  half  as  much  as  each  of  them  has  after 
the  first  division;  finally  C  pays  to  A  and  C  one-third  as 
much  as  each  of  them  has  after  the  second  division. 
Counting  their  shares  then,  A  has  $12,  B  $84,  and  C  $138. 
How  much  had  each  at  first  ? 

4.  A  and  B  together  can  do  a  job  in  12  days.  After 
they  get  it  three-quarters  done,  however,  they  call  on  0  to 
help  them,  and  thus  save  one  day.  C  can  do  as  much  work 
in  5  days  as  A  can  in  6.  How  long  for  each  alone  to  do 
the  entire  job  ? 

5.  A  and  B  together  can  do  a  certain  job  in  5  days;  if  A 
works  2  days,  B  3  days,  and  C  5  days,  y\  of  the  job  will  be 
done;  but  if  A  works  6  days  and  B  3  days,  the  job  will  be 
half  done.  How  long  will  it  take  each  alone  to  do  the 
entire  job  ? 

6.  A  number  consists  of  six  digits,  of  which  the  first 
is  1 ;  if  the  first  digit  is  erased  and  written  down  after  the 
other  five,  the  result  is  3  times  the  original  number.  Find 
the  number. 

7.  A  number  consists  of  three  digits;  their  sum  is  13, 
and  the  middle  digit  is  |  of  the  other  two;  if  297  be  added 
to  the  number,  the  digits  are  reversed. 


158  ALGEBRA. 

8.  If  5  kegs,  3  cans,  and  2  jars  of  oil  be  drawn  from  a 
full  cask,  the  cask  will  remain  \^  full ;  if  4  kegs,  5  cans, 
and  8  jars  be  drawn  from  the  full  cask,  the  cask  will 
remain  ^3_.  f^ii  J  and  if  2  jars  be  filled  from  a  full  keg, 
the  keg  will  then  contain  -gV  of  the  cask  full.  What  frac- 
tion of  the  cask  full  will  the  keg,  the  can,  and  the  jar 
respectively  contain  ? 

9.  A  farmer  received  $5745  for  horses,  cows,  and  sheep, 
at  prices  of  $110,  $62.50,  and  $7.50  apiece  respectively;  8 
more  sheep  would  have  brought  as  much  as  6  cows;  and 
the  total  number  of  animals  was  5  times  the  number  of 
cows.     Find  the  number  of  each. 

10.  A,  B,  and  C  subscribed  $100;  if  C  had  put  in  $2 
more,  and  B  ^^  more  than  he  did,  A  could  have  com- 
pleted the  sum  by  subscribing  -^^  less;  if  0  had  put  in 
$18.50  and  B  ^  more  than  he  did,  A  could  have  put  in  -J 
less.     Sum  each  subscribed  ? 


More  than  Three  Letters. 

174.  Equations  of  four  or  more  unknown  letters  are 
treated  in  a  similar  way.  From  the  original  set  of  equa- 
tions one  of  the  original  set  of  unknown  letters  is  elimi- 
nated, and  a  new  set  of  equations  is  formed,  with 

one  equation  less ; 

one  unknown  letter  less. 

This   is   repeated   until  we  have   one   equation   and   one 
unknown  letter ;  and  then  the  work  is  done. 

175.  Care  must  be  taken  that  each  equation  of  a  new 
set  is  derived  from  a  different  pair  of  the  old  equations, 
so  that  they  may  all  be  independent.  The  following 
example  shows  an  error  that  is  often  made. 


THE  FIRST  METHOD  OF  ELIMINATION, 


159 


Model  H. 

0  2a;  +  3y  --  2^;  +  bu  =  39 
®  Sic  -f  5i/  +  4^  -  3i^  :=  28 
(D  5^  -  2^  -  3;2  +  2t^  =  32 
®  4:^;  +  4y  —  5:2  —  4w  =  1 


Original  set. 


® 

4a;  +  6y  —  4z  +  10m 

=  78 

® 

X  2 

®   2«/  +  0  +  14w  =  77 

® 

-® 

® 

Qx  +  9y-  6«+15m 

=  117 

® 

X  3 

® 

6a;  +  lOy  +82  -  6m 

=  56 

® 

X  2 

'® 

-  y  -   142  +  21m 

=  61 

® 

-® 

@ 

13a;  +  20y  +  16«  - 

12m  = 

112 

® 

X  4 

@  12a;  +  l%y  -  15?  - 

12m  = 

3 

® 

X  3 

@ 

8y  +  3l2  = 

109 

@ 

-o 

® 

2y4-  2  +  14m  = 

77  1 

® 

—  y  —  14«  +  21m  = 

61  >•  New  set. 

® 

8.y  +  3l2  = 

109  ) 

@       6?/  +  3^  +  42w  =  231  ®  X  3 

@  -  2^/  -  28^+  42?/  =  122  (?)  X  2 

*  (15)  87/  +  31^  =  109  ®  -  ® 

Since  (j|)  and  @  are  identical,  the  three  equations  (§), 
(e),  and  @  are  kot  iN"DEPE]srDE]S'T.  Scanning  the  work 
we  find  that  (6)  was  obtained  by  combining  ®  and  Q,  (9) 
from  0  and  ©,  and  (jD  from  ®  and  0;  equation  (12)  there- 
fore contains  no  facts  that  are  not  already  in  ®  and  ®. 
And  if  these  three  equations  were  independent,  we  should 
have  determined  our  set  of  four  unknown  quantities  from 
three  equations, — which  is  impossible. 

176.  In  eliminating  with  several  unknown  letters,  then, 
one  must  be  careful  that  for  any  number  of  equations  of 
the  KEW  SET  at  least  one  more  of  the  equations  of  the 
original  set  has  been  utilized. 

In  the  work  given  above,  any  one  of  the  pairs  ®  0, 
0  0,  0  ©,  may  be  replaced  by  any  pair  containing  0. 


160 


ALGEBRA, 


Equation  (Tz),  obtained  from  (4)  @,  has  the  advantage  that 
it  can  be  used  without  change  as  one  of  the  next  new  set; 
it  would  be  better,  then,  to  put  some  other  pair  instead  of 
®  ®  or  0  @.  A  correct  solution  of  the  problem  would 
be  as  follows : 


Model  H. 

Q  2a;  +  3y  -  2^;  +  5t^  =  39 

(D  3a;  +  by  +  4.z  -  3^^  =  28 

®bx  -  2y  -  ^z  -\-  2u  =  32 

(a)  ix  -\-  4ty  ~  bz  —  4:11—1 

(5)  4a;  +  6y  -  4^  +  10u=  78 

*®      '^y  +  ^  +  l^^(^=   77 
®  lOo;  +  Iby  -  10^  +  2bu  =   195 
(8)  lOo;  —  Ay  —  Qz  +  4:U  =     64 

*  (?)    19y  -  4;^  +  21u  =   131 

@  \2x  +  20y  +  \^z  -  \2u  =   112 
(Q)  12a;  +  12y  -  15z  -  12^^  =  3 

*  @  8^  +  31^  =  109 


>  Original  set. 


®        2y  +  Z+UU 
®  19y  —  iz  +  2lu 
@Sy  +  31z  =  109 

=  77  ) 
=  131 

-  First  neii 

@    Qy  + 
®SSy  ~ 
©  32y  — 

3z  -}- 
8z  + 
Uz  = 

4.2u  : 

42W    : 

=  31 

=  231 

=  262 

@    8^  + 
@  32?/  - 

31^  = 

Uz  = 

=  109 
=     31 

t  Second  new  set. 

%  32«/  +  124«  =  436 
@      135^  =  405 

@  2=3 


©  X  2 

(D  -® 

®  X  5 
(D  X  2 

®-(D 

@  X  4 
®  X  3 
@-  ® 


®  X  3 
®  X  3 
(8)-(fD 


(iD  X  4 

(!D  -dD 

(17)  -^  135 


N.B.  It  is  convenient,  as  fast  as  tlie  equations  of  the  new  set  are 
obtained,  to  mark  them  with  an  asterislf  and  thus  save  perplexity  in 
looking  for  them. 


THE  FIRST  METHOD   OF  ELIMINATION.        161 

®  %  +  93  =  109  subst.  ®m® 

®1/  =  ^  [(19)  -  93]  -^  8 

@i  +  d  T^Uu  =  77  snbst.  ©and  @  in  (6) 

@)                      u=    5  [(2j)  -  7]  ■--  14 

@  2^  +  6  -  6  +  25  =  39  subst.  @,  @)  and  @  in  (?) 

@^  =  '7  [@  _  25]  ---  2 
ic  =  7;  ^  =r  2;  ;^  =  3;  w  =  5.     ^/^s. 

EXERCISE   LXXXVIII. 

1.  dx  +  2i/  +  z  +  6w=30  ^.  3x  -  2z  =  2 
2x-3y-5z  +  w  =  -l5  by-7io=  2 
x+  hy  -  3z  -  2w  ==-6  2^  +  3^  =  45 
bx  —  y  ~-2z-\-  'dio-  9  4^2  +  3z^  =  56 

2.  7iz;  -  5^  +  IO2;  -  2z^=  29  5.  2x  +  11;^  =  61 
^x-\-7y  -  bz  +  10w=  57  3w  -  Ux  =  -  12 
2:^;^+  3«/  +  72;  -  5w   =  16  10^  +  32:  =  59 
10:?;  _  2^  +  3^  +  7«^  :=  61  7u  +  13;2  =  83 

3.  2x+  5y  =  11  +  3z  +  2w    6.  ^x  -  3z  +  4:U  ~  23 

__  bz-2y  -  3w  6y  -  3z  +  2w  =  12 

^  ~        8  6«/  —  4a;  -  3i^  =  12 

22;  —  3.^  —  5«(;  5y  —  4«^  +  3?^  =  11 

^  8  4;?   -  9t(;  =  30 

^  =  1  +  i{3y  +  2w  -  8^) 

7.  2x-3y  +  4:z-15=z2y  -3z  +  4u-'7=^-^{2z-3u  +  4:x) 

=  M^  +  y  +  ^  +  ^)  =  ^ 

8.  3^-52;~2^  +  6r=10^  +  2;2-3?^H-4=:5^  +  3;2  — 8«(;  +  4 
=  13x  -  25«/  -  11?^  +  5  =  2;2 

9.  There  is  a  number  of  four  digits  such  that  if  the 
digits  are  reversed,  the  number  is  diminished  by  1089;  if 
the  second  and  third  digits  are  interchanged,  the  number 
is  diminished  by  90;  if  the  first  two  digits  are  removed  and 
placed  after  the  last  two,  the  number  is  increased  by  693 ; 
and  if  the  number  were  in  the  scale  of  9  instead  of  in  the 


16^  ALOEBRA, 

decimal  scale,  its  value  would  be  decreased  912.     Eind  the 
number. 

10.  A  traveller  has  a  large  number  of  foreign  bills,  all  of 
the  same  value,  also  gold,  silver,  and  copper  coins,  all 
coins  of  the  same  metal  having  the  same  value,  and  is 
vainly  attempting  to  pay  his  railroad  fare  with  them ;  he 
oifers  5  gold  coins,  6  silver  coins,  and  2  bills,  but  is  $1.10 
short;  then  he  tries  6  bills,  2  gold  coins,  7  silver  coins,  and 
6  copper  coins,  and  finds  that  makes  $4.03  too  much; 
6  gold  coins,  8  silver,  10  copper,  and  one  bill  is  again  $1.10 
short;  5  bills,  one  gold  coin,  and  24  silver  coins  make  $1 
too  much,  and  so  do  5  gold  coins,  20  silver,  100  copper, 
and  one  bill.  What  were  his  different  kinds  of  money 
worth,  and  what  was  his  railroad  fare  ? 


CHAPTER  VII. 
THE   SECOND   METHOD  OF  ELIMINATION. 

Linear  and  Quadratic  Pairs. 

177.  Model  A. — A  certain  rectangular  field,  containing 
480  square  rods,  requires  104  rods  of  fence  to  enclose  it. 
What  are  its  dimensions  ? 

Let  X  =  length  in  rods; 
then =  breadth  in  rods. 

X 

Q)    2x  +  2—  =:  104 

©       a;2  +  480  =  52:?;  Q  X  2;  -^  2 

(D  :z:2  __  52^  _|.  480  =  0  (2)  -  52a; 

and  so  on.     Or  otherwise. 

Let  X  =  length  in  rods; 
then  52  —  a;  =  breadth  in  rods. 
©  a:(52  -  x)  =  480 
©     52a;  —  x^  =  480  same  as  © 

®  a;2  -  52a;  +  480  =  0  ©  -  52a;  +  x^ 

and  so  on. 

Still  another  way  would  be  to  use  two  unknown  letters, 
as  follows: 

163 


164 


ALGEBRA, 


Let  X  —  length  in  rods;  y  =  breadth  in  rods. 


®  xy  =  480 

©  2x  +  2y  =  104 
(D     x  +  y    =52 


® 


Equations  Q  and  (3)  we  cannot  solve  by  the  method  of 
elimination  heretofore  tried.  We  can  find  the  value  of  x 
from  the  first  equation, — not  the  numerical  value  that  sat- 
isfies both  0  and  (3),  but  a  sort  of  formula  for  x, — and 
then  substitute  in  the  other  equation;  that  is,  write  instead 
of  X  the  formula  for  it.  Then,  if  the  formula  for  x  had  no 
letter  x  in  it,  we  shall  have  an  equation  from  which  x  has 
been  eliminated.     Thus, — 


®  X 


480 


©  480  +  /  =  52y 

(T)  y^  ~  52y  -f  480  =  0 

and  so  on.     Or  otherwise, 

(J)  X  =  62  -  y 

(5)  y{62  -  y)  =  480 

(6)  52y  -  y^  =  480 
(T)  0  =  y^  -  62y  +  480 


®-y 

(4)  substituted  in  (3) 

®  X  ^ 
®-  52y 


®-y 

(T)  substituted  in  0 

same  as  ® 

®  -  52^  +  y^ 


Or  we  could  in  the  same  way  have  eliminated  y]  and  the 
rest  of  the  solution  would  have  been  identical  with  the  so- 
lutions with  one  unknown  letter  which  were  worked  out 
first. 

The  use  of  the  second  unknown  letter  may  often,  as  in 
this  case,  be  a  way  of  explaining  how  to  get  with  one  letter 
abbreviations  for  the  two  unknown  numbers. 


TEE  SECOND  METHOD   OF  ELIMINATION,      165 

Model  B. — A  cistern  can  be  filled  by  two  pipes,  running 
together,  in  2  hours  55  minutes;  the  larger  pipe  by  itself 
will  fill  it  2  hours  sooner  than  the  smaller  pipe  by  itself. 
How  long  for  each  pipe  separately  ? 

X  =  number  of  hours  for  smaller  pipe; 

y  =  number  of  hours  for  larger  pipe. 

(J)  X  -  y  =  2 

®^  +  y  =  35 

(D  d6y  +  352;  =  12xy  ©  x  ^5xy 

®  x  =  y  +  2  (J)+y 

®  36y  +  36{y  +  2)  =  12y{y  +  2)     ®  subst.  in  ® 
(6)  12y^  +  24^  =  70y  +  70  same  as  ® 

0  12/  -  46y  -  70  =  0  (6)  -  70y  -  70 

(D     6/  -  23«/  -  35  =  0  Q)  -^  2 

(9)  (6^  +  7)  (2^  -  5)  =  0  ®  factored 

.-.     y  =:  5,  or  -  i 

5  hours  for  larger,  7  for  smaller.     Ans. 
Model  C. — A  number  of  two  digits  is  9  less  than  the 
square  of  the  sum  of  its  digits;  and  if  45  be  subtracted 
from  the  number,  the  digits  are  reversed. 

Let  X  =  the  tens  digit;  y  =  the  units  digit. 
®  10x  +  y  +  9  =  x^  +  2xy  +  if 
©  10:?;  +  y  -  45  =  10^  +  iC 

@  9^;  -  9y  =  45  @  —  lOy  —  a; 

®^-^  =  5  ®-^9 

®a;  =  ^  +  5  ®  +  y 

®  10(^  +  5)+y  +  9 

^^+10^+25+21/(^+5)+/   ©subst.  in® 
®  \\y  +  59  =  4/  +  2^y  +  25  same  as  ® 

®  4/  +  9;z/- 34  =  0  ®_lly-59 

®  (4y  +  17)(^  -  2)  =  0  ®  factored 

.-.   i/  =  2;    ^  =  -1^ 

iz;  =  ^  +  5  =  7  Ans,  72 


166  ALGEBRA, 

In  the  above  examples,  the  equations  of  the  first  degree,* 
since  they  may  be  represented  by  a  straight  line  in  a  dia- 
gram, are  called  linear  equations.  The  other  equation  is 
of  the  second  degree,*  and  is  called  a  quadratic.  The  rule 
for  this  method  of  elimination  (called  Elimination  by 
Substitution)  is : 

178.  Find  the  formula  for  x  (or  for  y)  from  the  linear 
equation,  and  substitute  in  the  quadratic. 

EXERCISE    LXXXIX. 


2. 

Q 

xy  :=^  1\     3x  —  6y  —  2. 

4. 
5. 

"^            2      -"^^  ^        x  +  2   -^' 
7x^  -  Sxy  =  159;       6x  +  2y  =  7. 
^2  _  2xy  -^  y^  =  31;  X  +  y    =  13, 

6.  3x  -  lOy  =  I;  x^  --  xy  =  by^  +  79. 

7.  X  +  y  =  7;  x^  +  2y^  =  34. 
S.  X  =  y  -  2;  3x^  =  4y^  +  48. 

9.  ^  -  2x  =  Q;  4:X^  =  9{y^  +   12). 

10.  x^  +  24.xy  =  {3y  +  4.y  +  S;  x  =  3y  +  2. 

11.  -  +  |  =  1;  -  +  -=4. 
2   3     X   ^  y 

12.  3x^  -  2xy  =  15;  2x  +  3y  =   12. 

14.  11«/  +  6x  =z   63;  x^  -  xy  +  f  =   13. 
16.  ^-^-'^~f  =  l;x  +  y  +  3xy^93. 

*  Tlie  degree  of  an  equation  is  the  degree  of  its  highest  term  ;  and 
the  degree  of  a  term  is  the  number  of  unknown  letters  that  are  fac- 
tors of  it ;  thus  dx^y^  would  be  of  the  fifth  degree,  because  it  con- 
tains two  factors  x  and  three  factors  y.     See  §  75. 


THE  SECOND  METHOD   OF  ELIMINATION.      167 

16.  ^^^-^^  =  ^;  a;2+6a;«/  +  2«/'  +  4x  +  3«/  =  29. 
n.   ]0-  ?^±^  =  ^;  3x^  -  6=.  +  52/  =  20. 


18. 


X  -  y  -  1  y 

20.   ?^±|-^  ^  ^+_^;  .V^  +  4r.^  :^  ^f  +  20:.  +  6. 

21.  Find  two  numbers  whose  sum  is  4  times  their  differ- 
ence, and  the  difference  of  whose  squares  is  196. 

22.  A  path  around  a  rectangular  garden  is  7  feet  wide 
and  1806  feet  in  area,  which  is  994  feet  less  than  the  area 
of  the  garden  itself.     Find  the  size  of  the  garden. 

23.  A  number  is  7  times  the  sum  of  its  two  digits;  and 
if  the  number  is  multiplied  by  its  first  digit,  the  product 
is  672.     Find  the  number. 

24.  The  difference  of  two  numbers  is  -^^  of  the  greater, 
and  the  difference  of  their  squares  is  380.  Find  the 
numbers. 

25.  If  a  certain  rectangular  field  were  75  rods  longer 
and  20  rods  wider,  its  length  would  be  double  its  breadth, 
and  its  area  would  be  double  what  it  is  now.  Find  its 
dimensions. 

26.  There  are  in  a  certain  block  of  exactly  similar 
houses  300  rooms;  5  more  houses  in  the  block  than  there 
are  rooms  in  one  of  the  houses.  How  many  houses  in  the 
block  ? 

27.  The  front  wheel  of  a  bicycle  makes  16  turns  less 
than  the  hind  wheel  in  going  a  mile;  if  the  front  wheel 
were  6  inches  more  in  circumference,  it  would  turn  60  times 
less  than  the  hind  wheel  in  going  a  mile.  Find  the  cir- 
cumference of  each  wheel. 


168  ALGEBRA. 

28.  When  a  certain  train  has  travelled  5  hours  it  is  still 
60  miles  short  of  its  terminus;  and  on  the  whole  trip  1 
hour  can  be  saved  by  running  5  miles  an  hour  faster. 
Find  speed  of  train  and  length  of  trip. 

29.  A  and  B  start  on  a  road  race  together;  A  is  a  sure 
winner,  and  looking  back  once  on  the  road  he  sees  B  60 
rods  behind.  A  crosses  the  line  4  minutes  after  that,  and 
B  comes  in  9  minutes  behind.  When  A  looked  back  he 
had  as  far  to  ride  as  B  had  already  ridden.  Find  their 
speeds. 

30.  The  fore  wheel  of  a  carriage  turns  132  times  more 
than  the  hind  wheel  in  going  a  mile;  and  6  turns  of  the 
fore  wheel  cover  2  feet  less  than  5  turns  of  the  hind  wheel. 
Circumferences  ? 

31.  Kowed  24  miles  down-stream  and  back  again;  took 
8  hours  longer  on  the  return  trip,  the  current  reducing 
the  speed  to  -J  of  what  it  was  going  down.  Kate  of  the 
current  ? 

32.  Two  trains  start  from  opposite  ends  of  a  double- 
track  railroad  300  miles  long;  after  they  pass,  one  train 
takes  9  hours,  the  other  4,  to  complete  the  journey.  Speed 
of  each  train  ? 

33.  If  1  is  added  to  the  denominator  of  a  fraction,  the 
value  of  the  fraction  becomes  J;  and  if  2.1  be  added  to  the 
fraction,  the  fraction  is  inverted.     Find  the  fraction. 

34.  If  2  be  added  to  the  denominator  of  a  certain  frac- 
tion, its  value  becomes  J;  if  4  be  added  to  each  term  of  the 
fraction,  its  value  is  increased  by  ^.     Find  the  fraction. 

35.  A  rectangular  field  238  rods  in  area  loses  58  square 
rods  by  taking  off  a  strip  1  rod  wide  all  around  the  edge 
for  roads.     Find  the  dimensions  of  the  field  that  remains. 

36.  Five  leaks  and  a  drain-pipe  empty  a  cistern  in  4 
hours ;  the  average  time  for  one  of  the  leaks  to  empty  it 
would  be  24  hours  more  than  half  the  time  the  drain-pipe 


THE  SECOND  METHOD    OF  ELIMINATION,      169 

requires.     How  long  will  it  take  the  cistern  to  be  emptied 
if  the  leaks  are  stopped  ? 

37.  When  a  certain  kind  of  cloth  is  wetted  it  shrinks  up 
^  in  length  and  -j^  in  width.  A  piece  of  this  cloth  used 
for  an  awning  was  found  to  have  shrunk  8|  square  yards 
in  area,  while  the  edging  bought  for  it  before  it  shrank, 
intended  to  go  on  one  long  side  and  both  ends,  was  found 
to  be  2  yards  too  long.  What  was  the  length  and  the  width 
of  the  cloth  before  shrinking  ? 

38.  The  area  of  a  rectangular  plot  of  land  is  16  feet  less 
than  a  square  plot  of  equal  perimeter;  and  its  breadth  is 
one  foot  more  than  |  of  the  length.  Find  the  dimensions 
of  the  rectangular  plot. 

39.  Invested  in  5,^  stocks;  then  invested  the  same  sum 
in  6^  stocks,  paying  $6  higher  premium  and  getting  2 
shares  less;  the  income  from  the  second  investment  is 
$18  more  than  that  from  the  first.  What  was  the  sum 
invested  ? 

40.  Two  boys  run  in  opposite  directions  around  a  rect- 
angular field  the  area  of  which  is  one  acre;  they  start  at 
one  corner  and  meet  13  yards  from  the  opposite  corner; 
and  one  of  the  boys  could  go  around  the  field  6  times 
while  the  other  is  going  5  times.  Find  the  dimensions  of 
the  field. 


PAIRS  OF   QUADRATICS. 

179.  If  we  were  to  make  diagrams  for  the  equations  in 
this  chapter  as  we  did  for  some  of  the  equations  in  the 
preceding  chapter,  we  should  find  that  the  points  for  an 
equation  of  the  second  degree  formed  a  curved  line, — a 
circle  or  an  ellipse,  or  perhaps  one  of  the  two  kinds  of 
curves  shown  in  Fig.  8, — while  the  points  for  an  equation 
of  the  first  degree  formed  a  straight  line. 


170  ALGEBRA. 

180.  A  straight  line  can  cut  one  of  these  curves  twice; 
that  is,  there  are  two  points  which  represent  an  answer  to 
both  equations,  and  consequently  we  always  expect  two 
answers  when  we  have  one  equation  of  the  first  degree  and 
one  of  the  second.  When  both  equations  are  of  the  second 
degree  the  case  is  different:  then  both  of  the  lines  which 
represent  the  equations  are  curved  lines,  and  they  can  cross 

"  Parabola." 
"  Hyperbola." 


Fig.  8. 

each  other  four  times,  as  in  Fig.  7,  Chapter  VI.  In  other 
words,  two  equations  of  the  second  degree  have  four 
answers, — four  sets  of  particular  values  for  x  and  y  that 
satisfy  both  of  the  given  equations. 

181.  This  very  fact, — that  there  are  four  answers,  and 
consequently  that  the  equation  we  should  get  after  sub- 
stituting would  have  four  factors, — is  enough  to  show  us 
that  we  cannot  generally  solve  a  pair  of  quadratic  equa- 
tions, because  we  cannot  generally  factor  an  equation  of 
the  fourth  degree. 

182.  Special  cases  arise,  however,  which  can  be  solved 
by  some  special  method, — sometimes  requiring  a  great  deal 
of  ingenuity. 

Only  Two  Kinds  of  Terms. 

183.  In  the  following  set  of  examples,  since  there  are 
only  two  kinds  of  terms  that  are  not  similar,  the  method  of 
combination  may  be  used : 


THE  SECOND  METHOD   OF  ELIMINATION,      171 


EXERCISE  XC. 

^'  (    :?;2  _|_  2?/2  ^57  ^-  (  3^2  _    ^2    ^  ^^ 

(  3:^:2  _  5^^^     7  ^4    (3:^^+57/'^    =155 

•\4.xy-  22;'^  =14  '  (  V  -  ^^^y  =  15 


# 


6. 


j  15:?;2  _|_  4^^  ^  40 
(  "ixy  -  12:^2  =  2 


Finding  a  Linear  Equation. 

184.  The  first  effort  of  a  pupil,  in  trying  to  solve  a  pair 
of  quadratics,  should  be  to  work  out  from  them,  if  possible, 
some  equation  of  the  first  degree,  which,  with  either  of  the 
original  quadratics,  will  form  a  Lii^EAR  akd  quadratic 

PAIR. 

Model  D. 

®  2^2  _  5^  .^    3^  ^  171 

®  3:z;2  ^_  3^  _  13^  :^  239 

(D  6:^2  _  15^  +  9y  =  513  ®  X  3 

®  62^2  +  6:^;  -  26^  =:^  478  @  X  2 

(5)  21^  -  35^  =  35  ®  "-  (D 

(6)3:c-5^=-5  ®^7 

®  2/  =  -^^  [(D  +  5  +  52/]  -  5 

®  2:z;2  _  5:c  +  3  — ^  =  171        ®  substituted  in  ® 

®  10.^2  -  25:^;  +  9ic  +  15  =  855     ®  X  5 
@  10r7;2  _  16:?;  -  840  =  0  ®  -  855 

(0)    5:^2  -    8a;  -  420  =  0  ®  -r-  2 

@  (5a;  +  42)(a;  -  10)  =  0 
.  •.  ic  =  10  or  —  4^3- 

*  See  footnote,  p.  173. 


172 


ALGEBRA, 


In  this  example  we  can  get  only  one  equation  of  the  first 
degree  to  substitute  from ;  and  consequently  here  (that  is, 
where  the  terms  of  the  second  degree  are  alike  in  any  pair 
of  quadratics  *)  we  can  get  only  two  sets  of  answers. 

The  values  of  y  are  obtained  by  substituting  the  values 
of  X  successively  in  the  equation  of  the  first  degree  by  which 
they  were  obtained  (in  this  case  (f)). 


5y=-5 

126   , 
5    %=   5 

5^=35 

126  +  25?/  =  25 

25«/  =  -  101 

101 

^=   25 

185.  In  order  to  show  what  values  of  x  and  y  belong  to 
the  same  answer,  it  is  best  to  arrange  the  results  in  a  table, 
thus : 


X 

10 

-¥ 

y 

7 

-Y/ 

(D  X  13 

©  X  3 

@  +  ® 

®-4-7  - 

420 

186.  We  might  also  have  solved  by  eliminating  y: 

(D  26:^2  -  Qbx  +  39^  =  2223 
®  9a;2  +  9x  -  39^  =  717 
®  ^bx^  -  b^x  =  2940 
(6)  bx^  -  8^  -  420  =  0 

whence  as  before. 

This  method  can  be  used  when  in  both  equations  all  the 
terms  containing  one  of  the  unknown  letters  are  similar. 

*  In  fact,  whenever  tlie  expressions  made  up  of  terms  of  tlie 
second  degree  in  each  equation  have  a  common  factor  containing  x 
or  y. 


THE  SECOND  METHOD   OF  ELIMINATION,      173 


One  Quadratic  Factorable. 

187.  Sometimes  one  of  the  given  equations  can  be 
factored : 

Model  E. 

0  f  —  ^xy  +  8i/  ==  16a;  —  7  —  ^x^ 
(D  x^  -f  2xy  =  24 

(D  4:X^-4:xy+y^-16x-irSy  +  '7=0  (T)  +  7+4:X^-16x 

®  {2x  —  yf  —  8(2a:  —  y) -\-  7=0  same  as  (3) 

(5)  (2:r  -  ^  -  7)(2a;  -  y  -  1)  =0  ®  factored 

(^^x-y-l=:0  (5)  Ax.  A 

®^z=2a;-7  ®  +  y 

®2x-y  —  1  =  0  (5)  Ax.  A 

®y  =  2^-l  ®  +  ^ 

(10)  ^2  _|_  2a;(2:r  -  7)  =  24  ®  subst.  in  @ 

^  5^2  __  14^  _  24   =0  ®  ~  24 

@  {bx  +  io){x  -  ^)  =  0  @  factored 

.*.  .7;  =  —  I;     :r  =  4 

(g)  2:2  +  2^(22;  -  1)  =  24  ®  subst.  in  @ 

@  52;2  -  2a;  -  24     =  0  @  -  24 

@  {bx  -  12)  (a;  +  2)  =  0  @  factored 

a;  =  -igS;  :^;  =  —  2 

Substituting  these  four  values  of  x,  EACH  IN  THE  LINEAE 
EQUATION  BY  WHICH  IT  WAS  OBTAINED,  we  get  the  fol- 
lowing four  answers : 


X 

-  1 

4 

V- 

-  2 

■  y 

-¥ 

1 

-V- 

-  5 

Of  these  four  answers  the  first  two  depend  on  ®,  and 
the  others  on  ®. 


174 


ALGEBRA. 


One  Quadratic  Homogeneous. 

188.  The  method  just  described  can  be  applied  when- 
ever one  equation  of  the  pair  of  quadratics  is  homogeneous, 
that  is,  where  every  term  is  of  the  same  degree.  A  homo- 
geneous equation  with  two  unknown  letters  is  also  called  a 
binary  equation. 

Model  F. 

©  x^  -\-  xy  +  2«/2  =  74 
(D  24«/2  -  "Zbxy  =  2bx^ 


(3)  25x^  +  25xy  ~  24^^  _  q 
®{bx  +  Sy){bx-^)  =  0 
®bx-\-Syz=zO 
6x 


®y  = 


8 


^_  6x 

®^-¥+^  =  « 

@7ix^  =  74  X  64 
Qx^  -  64:  =  0 

.\  X  =  S;  X  =  —  S 

®  ^'  +  T  +  ¥  ==  ^4 
@  74.x^  =  74  X  9 
@x^  -  9  =  0 

X  =  d;  X  =  —  3 


©  +  25xy  —  24^2 
@  factored 
©Ax.  A 

[®  -  5:^]  H-  8 

©Ax.  A 

[®  +  3^]  ^  3 

©  substituted  in  © 

®  X  64 

@  -^  74  -  64 

©Ax.  A 

®  substituted  in  © 

(JD  X  9 

@  -^  74  -  9 


Eemembering  that  the  first  two  values  of  x  were  found 
on  the  supposition  that  y  =  —  -^-,  and  the  others  on  ®,  we 
get  for  our  table  of  answers 


THE  SECOND  METHOD   OF  ELIMINATION,      176 


X 

8 

-  8 

3 

-  3 

y 

-  5 

5 

5 

-  5 

189.  The  general  principle  of  finding  a  linear  equation 
from  the  pair  of  quadratics,  either  by  eliminating  one  kind 
of  terms  or  by  factoring  one  of  the  equations,  is  illustrated 
by  the  following  examples. 


EXERCISE  XCI. 


Find  four  answers  wherever  possible : 
^  ^Zx^  -2x-\-  bxy  =  96     ^   {x^  -  ^0  =  2y2  +  2\y 

(  a;2  -  IO2;  +  25  =  4^2 


(  ^xy  —  5^  =  15 

3.  [  ^^' 

Vlx' 


^x^  -  Sxy  +  dy^  -8y  +  29=0 


2x^  -{-  y^  =  4:xy  —  7 


^   (  3x^  -  7xy  +  y^  +  9y  +  Q6  =  0 
'i  3x^  +  28y^  +  4:8  =  7xy 
(  x^  -{-  xy  -\-  y^  =  19  r  i  ^^  "^  ^^^  ~  ^^ 

'\6x^-  Uxy  +6y^=:  0  (  lUx^+  291xy  =  216y^ 


+  5xy  —  6y^  =  0 


(xy  +  M  =  0 

I  Qx^  +  Idxy  +  6y' 


M 


xy  =  y 
x^  -f  ^^ 


0 
2  _  4 


•■i 


2^2  +  27/  =  21a;^ 
x^  +  Zxy  =  27 


3xy 


11.  18a;2  -  9xy  =  Ux^  +  28/;  x^  +  2/  =  18. 

12.  IQx^  +  Sxy  =  16x^  —  15/;  2x^  -\-  xy  =:  15. 

13.  24:r'^  +  Idxy  =  2/;  x^  -{-  xy  —  y^  =  —  11. 

14.  4:x^  +  12/  =  7x'  +  Ixy  +  14/;  x^  +  3/  =  28. 

15.  69a;2  +  23^;^  -  69/  =  660;'-^  -  33/;   ^x'^  -  /  =  23. 

16.  ^x^^^Oxy-^y'^  =lx^+21xy  -  14/;  2/-a;2-  3x^=4. 

17.  Iliz;2-ll:r2/-176/=2a;2+22:y-16/;  a;2+^«/=22+8/. 

18.  5^^  +  5/  =  29x^  —  58^^:1/ >  ^^  ~  ^^^  ^  ^' 


176  ALOEBEA, 

19.  —  3a;2  4-  Qxy  —  d2xy  +  l^y^]  x^  —  Ixy  =  16. 

20.  l^x^  +  45^^  =  —  *7xy  —  28y^;  xy  +  4^/^  =  30. 

190.  The  two  equations  obtained  by  factoring  a  binary 
equation  are  either  inconsistent  or  identical;  and  further, 
two  binary  equations  can  never  be  simultaneous,  for  the 
same  reason.     Find  out  ivhy. 

Finding  a  Binary  Equation. 

191.  When  in  each  equation  the  terms  containing  the 
unknown  letters  are  all  of  the  second  degree,  it  is  possible 
to  get  a  homogeneous  equation  by  eliminating  the  numeri- 
cal terms : 

Model  G. 

®  ^2    _|_  ^2   ^    5 


®  Sx^  +  8^2  ^  40 

®  X  8 

®    IQx^ -{•  bxy -\- ^y^  =  ^^ 

@  X  5 

®  2x^  +  6xy  -  dy^  =  0 

®-® 

®  {2x-y){x  +  3y)  =0 

®  factored 

(T)                   2x  —  y  =z  0 

®  Ax.  A 

®  y  =  2x 

Q)  +  y 

®  X  +  3y  =  0 

®  Ax.  A 

®y=-l 

[®  -  ^]  -^  3 

([})  x^  +  4.x^  =  5 

®  substituted  in  ® 

(ID  :?;2  _  1  ^  0 

@-  5  -1 

@  {x+  l){x  -  1)  =  0 

@  factored 

®x  +  l  =  0.'.x=-l 
©aj  —  1  =  0.-.  ^  =  1 

1  (13)  Ax.  A 

@                x^  +  ^-  =  5 

@  substituted  in  ® 

@  10^'-^  =  45 

@  X  9 

@  a;2  ~  1  =  0 

dD-io-i 

THE  SECOND  METHOD   OF  ELIMINATION,      177 

Since  the  square  root  of  f  =  2.121+^  we  can  continue 
the  work  as  follows : 

@  (^  +  2.121)  (a;  —  2.121)  =  0  @  factored 
@  a; +2.121=0;  a;  =  ~  2.121  ®  Ax.  A 
©a; -2.121  =  0;  a;  =  2.121     (I9)  Ax.  A 

192.  The  values  of  x  given  in  @  and  @  were  obtained 
on  the  supposition  that  ®  was  true,  that  is,  that  y  =  2x; 
and  the  other  values  of  x  were  obtained  on  the  supposition 

X 

that  @  was  true,  that  is,  that  y  =  —  - ;  consequently,  we 

o 

must  be  careful  to  substitute  (u)  and  (15)  in  (8)  [not  in  one 

of  the  original  equations],  and  @  and  @  in  @.     In  this 

way  we  get 


X 

1 

-  1 

2.121 

-  2.121 

y 

2 

-  2 

-  .707 

.707 

Instead  of  figuring  out  the  square  root  of  f  we  might 
have  represented  it  by  the  symbol  V|;  our  table  of  answers 
would  then  have  been  ;        :  \ 


X 

1 

-  1 

VI 

r% 

y 

2 

-  2 

-iV% 

\v% 

193.  Whenever,  as  in  this  case,  the  root  of  a  number 
cannot  be  exactly  expressed,  the  symbol  that  stands  for  its 
exact  value  is  called  a  surd,  or  irrational  quantity.  Later 
we  shall  learn  methods  of  reducing  and  combining  such 
quantities. 


178  ALGEBRA. 


EXERCISE  XCII. 


1 


i    x^  +  xy  '\'  y'^=2S  {  2x'  J^  xy  +  x  =  21 

I  2:z;y  +  y2  _  ^2  ^  28  '  (    9^'^  +  2x^  ==  ^xy 

a;2  +  a;^  =  27  ^^'  (  ^^  4,  3^2  ^  28 

:?;2  +  Uy  +  2^/2  =  40  3  I  ^^  +  V  =  32 

2a;2  ~  2a;y  +  y2  ^  5  ^  '  (  3:r?/  +  6/  -  4  =  32 

C  a;2  +  9a:y  =  340  C  15^^  _  ^xy  =   111 

I  7a;^  -  y^  =  171  '  \  12(xy-  /)  r=  3x^  -  :r.v 

^^  +  ^y  +  2y^  =74  j  3a;y  +  6y2  =  4:y  +  4cO 

2x^  -\- 2xy  +  y^  =^  73  '  \     x^  +  6y^  =  6xy 

(bx^  +  3xy  +  2y^  =  lS8  (  x^  +  3xy  :=  4.0 

'  \      x^  -  xy  +  y^  =  19  {x^  +   3/  =  28 

^  (Sx^  —  3xy  -  y^  =  4:0  (  6x^  +  31xy  —  Iby'^  =  QQ 

'  \9x^  +  xy  +  2t/  =60  '^'  \  5x^  -  y^  =  5 

(      x^  -  xy  +  y^=7  (2x^  -  xy  -  6y^  =  23 

*•  I  3x^  +  13xy  +8^2^:,:  162  ^®'  \xy  =  21 

i2x^-\-  3xy   =  26  r  2a;2  +  l\xy  =   120 

(  3«/2  +  2a:^  =  39  ^  •  I  a;2  ~  30^-^  =  105 

10.  i^  -  '?  =  !J  20.  i  ''"^/''^loV+'^) 


THE  SECOND  METHOD  OF  ELIMINATION.      l79 


The  "Symmetrical"  Method. 

194.  Where  both  equations  are  symmetrical, — that  is, 
where  x  and  y  enter  into  each  in  such  a  way  that  if  they 
were  interchanged  the  resulting  equation  would  be  identi- 
cal with  the  given  one, — a  method  of  combination  is  em- 
ployed to  reduce  the  two  equations  to  the  form  x-\-y  =  ,  ,  . 
and  a;  —  2/  =  .  .  .,  as  illustrated  by  the  following  examples: 

Model  H. 


©  a;  +  y  =  15 

©        xy  =  36 

®  x^  +  %xy  ■\-  y^  =  225 

®' 

®                     ^xy  =  144 

®  X4 

(D  a;2  -  Zxy  +  j,^  =  81 

®-® 

@{x~  yY  -  81  =  0 

®  -  81 

(J)x-y  =  9;     ®x-  y 

=  -  9     ®  Ax. 

(D  2a;  =  24           ®  +  ® 

@  2y  =  6             ®  -  ® 

©  2a;  =  6             ®  +  ® 

©  2y  =  24           ®  -  ® 

X 

12 

3 

y 

3 

12 

Ans, 


Model  I. 

®x^  +  y^  =  74 
0  xy  =  35 


(D  x^  +  2xy  +  y^  =  144 
®x^  -  2xy  +  y^  =  4: 
®  {x  +  yY  -  144  =  0 


©  +  2  X® 
©  -  2  X© 
(D  -  144 


180 


ALGEBRA. 


® 
® 
® 
(Q) 

® 
® 
(1^ 


(a;  _  y)»  _  4  =  0  ®  -  4 


x  +  y  =  12; 
X  —  y  =  %; 
2x  =  14 
2y  =  10 


2a;  = 
2y  = 
2x  =  10 
14 


10 

14 


®2y 

@2x  =  -  U 

@2y=  -10 


(S)x  +  y  = 
@x  —  y  = 
®  +  ® 
®-® 
(D  +  ® 
®-® 
® +  ® 
®-  ® 
® +  ® 
®  -  ® 


-  12 

-  2 


(D  Ax.  A 
(6)  Ax.  A 


X 

7 

-  5 

5 

-7 

y 

5 

-  7 

^  7 

-  5 

Ans, 


Model  J. 

(T)  a;'  +  y''  =  185 

®    a;  +  y=17 

(D  a;'  +  2a;y  +  ^^  =  389 

©^ 

0                     2xy  =  104 

®-® 

®x^  —  2xy  +  y»  =  81 

®-® 

®  (a;  -  2/)^  -  81  =  0 

®  -81 

®a;  -2/  =  9;     (f) «  -  ^ 

=  -  9     ®  Ax.  A 

®  2a;  =  36 

® +  ® 

®  2y  =  8 

®-® 

(Q)3a;  =  8 

© +  ® 

@  22/  =  36 

©  -  ® 

,     X 

13 

4 

y 

4 

13 

Ans, 


THE  SECOND  METHOD   OF  ELIMINATION,      181 


195.  Sometimes  the  equations  are  symmetrical  except 
that  the  interchange  of  variables  produces  a  change  of 
sign.  The  general  method  illustrated  above  is  still  appli- 
cable. The  same  interchange  can  of  course  be  made  in 
the  set  of  simple  equations  that  constitute  the  answers; 
and,  consequently,  in  these  as  in  all  examples  that  can  be 
solved  by  the  symmetrical  method,  it  is  only  necessary  to 
find  half  the  answers, — the  others  can  be  written  down  by 
making  the  interchange  of  x  and  y. 

Model  K. 

©  a;  -  2/  =  3 
(2)  a;2  +  y'^  =  65 


(^  x^  -  2xy  -\- y^  =  ^ 

& 

©  %xy  =  56 

®-(D 

(D  a;«  +  2xy  +  y^  =  121 

®  +  ® 

(D  («  +  yf  -  121  =  0 

®  -121 

Ox  +  y  =  11;  ®x^y  = 

-  11 

®  Ax.  A 

CD  2a;  =  14 

®  +  ® 

@  2«/  =    8 

®-® 

(0)  3ar  ^  -  8 

®+® 

©  2y  =  -  14 

®-  ( 

x» 

X 

7 

-  4 

A 

y 

4 

■  -  7 

xi 

A  Short  Cut. 

196.  When  a  quadratic  equation  is  prepared  for  factor- 
ing, if  the  first  straight  product  is  oc^^  the  second  straight 
product  is  the  product  or  the  two  roots;  and  the  sum 
of  the  cross  products  will  be  —  a;  multiplied  by  the  sum  of 
THE  ROOTS.*  For  instance,  the  equation  iz;^  —  7a;  +  12  =  0 
has  answers  x  =  'd  and  x  —  4=. 

*  See  footnote,  p.  124. 


182 


ALGEBRA, 


Now  this  property  of  the  quadratic  equation  can  be  made 
use  of  in  solving  symmetrical  equations;  the  values  of  cry 
and  X  -\-  y^  or  of  xy  and  x  —  y,  are  early  obtained;  and 
remembering  in  what  cases  the  two  values  of  x  are  opposite 
in  sign,  we  shall  know  when  to  make  the  second  straight 
product  negative: 
Model  L. 

(\)x-\-y  =  n 
@        xy  z=z  ^% 

Auxiliary  quadratic, 
©Ax.  A 


®  o;^  -  13a;  +  36  =  0 
®  a;  =  9;  a;  =  4 


X 

9 

4 

y 

4 

9 

y=^l 


(3)  Ax.  A 


Model  M. 

0a;  —  ^  =  7  or  a; 
®xy  =  1S 

@x^^lx  —  lS  =  0  or  x'^  +7a;-18=0     Aux.  quad. 
®a;=9        I  a:=-9 

a;  =  -  2  i   ^^*    x  =  2 


Ans. 


197.  The  auxiliary  quadratic  constructed  in  this  way  is 
precisely  the  same  as  would  be  obtained  by  the  method  of 
substitution. 

Irregular  Devices. 

198.  Equations  of  higher  degrees  than  quadratics  can 
sometimes  be  solved.  Often  considerable  ingenuity  is 
required  to  get  a  solution;   and  it  must  be  remembered 


X 

9          -  2 

-  9 

2 

y 

2          -  9 

9 

9 

THE  SECOND  METHOD   OF  ELIMINATION.      183 

that  it  is  only  an  accident  when  a  pair,  even  of  quadratics, 
is  solvable  at  all  by  elementary  algebra. 

One  device  of  frequent  use  is  to  divide  one  equation  by 
the  other. 

Another  device  that  is  often  of  great  service  may  be 
applied  to  example  14,  below.  Let  z  stand  for  x^  -\-  y^  and 
u  for  ^xy.  The  two  equations  then  become  3;^  —  w  =  27 ; 
4;2  —  3w  =  16.  When  u  and  z  are  found,  equate  their 
values  to  '%xy  and  x^  +  y^  respectively  and  the  rest  of  the 
way  is  familiar. 

EXERCISE  XCIII. 

In  the  folloiving  examples  occasional  hints  are  given  to 
enable  the  student  to  choose  a  good  method  of  elimination  : 
IX  —  y    =    8;  a:y  =  33. 
2.  x^  -\-y^  =  68;  xy  —  16. 
3    ^3  _  ^3  ^  37.  ^z  ^xy  -\-y^    =  37. 

4.  x^  -{-  y^  =  152 ;  ay^  —  xy  -}-  y^  —  19. 

5.  a;3  +  ^3^  407;  x-\-y^\\. 

6.  x^  -y^  =  2197;  :r  -  y  =  13. 

7.  x^  +  xY  +  y^  =  2128;  x^  +  xy  +  y^  =  76. 

,,^-  +  l  =  2i;x  +  y  =  6. 

34  15         .  „ 

9.     ^  ,      2  =  — ;  X  +  y  =  S. 
x^  +  .V      ^y 

10.  x^  -{-  xy  =  i2;  X  -{-  y  =  5, 

11.  2x  -{-  y  =  x^;  2y  -\-  x  —  y^,     [Subtract  and  factor.] 

12.  x^  +  xy  =  210;  if  +  xy  =  231.     [Add.] 

13.  xy{x  -\-  y)  =^0]  x^  -{-  y^  =  35.     [Divide  and  cancel.] 

14.  3(2;^  +  y^)  -  2xy  =  27 ;  M.x^  +  f)  -  ^xy  =  16. 

[Solve  for  x^  -}"  ^^  ^^^  ^^'^O 

15.  :c^  +  4:xy  -f  ^3  _  33.  ^  _|_  ^  _  2.     [Divide.] 

X  -X-  y  8 

16.  -y-^  =  x  +  y  +  V  ^^  ^  ■'^^*    tSolve  first  for  x  +  y,] 


184 


ALGEBRA. 


17.  a^  +  y^  =  —^  ]  ^  +  y 


9.     [Divide.] 


18.  x^  +  y^  =  -^;  ^  +  y  =  ^ 

[®  -^  ®  —  ©^;  solve  for  a;?/  and  substitute.] 

19.  x^  -\-  y'^  =  xy  -\-l^',  X  -{-  y  =  xy  —  6. 

[®  +  ^^y  5  solve  for  xy  and  a;  +  3/'] 

20.  ^'^  +  ?y2  =  74;  10(:z;  +  2/)  =  ^^^^  -  ^• 

[®  +  f®;  solve  for  a;  +  ^.] 

Symmetry  not  Obvious. 

199.  The  following  examples  may  be  solved  by  the  sym- 
metrical method  ;  for  though  they  are  KOT  symmetrical  in 
X  and  y,  they  are  symmetrical  in  functions*  of  x  and  y ;  just 
as  the  equations  of  Model  M  are  symmetrical  in  x  and  —  y. 
For  convenience  we  may  adopt  the  same  device  as  in  example 
14  above,  letting  z  and  w  stand  for  the  functions  of  x  and  y, 

27^3     I     ^.6 

Model  N.     9x^  •-- dxy' +  y' =  ^  ^ 

Let  z  =  dx;  w 


10 


28. 


y'^\  then  the  equations  become 

z^  —  z\o  4-  w^  r=  28 
^3  _j_  ^3  ^  280 

whence  we  obtain  ;^  =  6  or  4  ;  «^  =  4  or  6. 

\iZx^^,x  =  2;  If  ?/2  =  4,  ^  =r  ±  2;__ 

if  3a;  =  4,  :z;  =  |.  if  /  =  6,  ?/  =  ±   1^6. 


X 

2 

2 

i 

1 

y 

2 

-  2 

VI 

-  |/6 

*  A  FUNCTION  of  X  is  an  expression  containing  ic;  e.g.,  2iC  —  1, 

3x2  _^  7^  etc. 


THE  SECOND  METHOD  OF  ELIMINATION,      185 


EXERCISE    XCIV. 


■•{ 


^x^  +  9«/2  ==  181 
xy  =    15 

x^  +  "^xy^  +  %4  ^  273 
:?;3  _  %y^  =  819 

x^  +  9y^  =  873 

'-  +  '-  =  7 

X       y 

xy  =  1 
[Let  z  =  3x;  w  =  4«^.] 


6. 


■x-\-y'^  =  60 
^2  +  y  =  2522 

7.  •<  3:?;       oy 
(6x+  5y  =  4: 

/  i^;2  _j_  4^2  ^  4^  _|.  99 

8.  •)  2a:^  —  x  =  4:S 

(      [Let;2  =  2?/ -  1.] 

r  a;^  +  9^2  -  i2y  r=  9 

9.  -I  ^xy  —  2x--Q 
{      [Let  2;  =  3^  -  2.] 


10.  < 


Let  2;  =  - ;  w  =  3y^. 


4a;H  9^/2+122^-6^=175 
2x  —  9y  —  6xy  =  85 
[Let  z  =  2x  +  ^] 
w  =  3y  -  1] 


Elimination  by  Comparison. 

200.  Still  a  third  method  of  elimination,  called  ^^com- 
parison," is  recognized  by  students  of  algebra,  and  is  some- 
times of  great  convenience,  though  it  is  never  necessary. 
It  consists  in  finding  an  expression  (a  ^'  formula ")  for  y 
(or  for  x)  from  each  equation  and  setting  them  equal. 
Model  0. 

Q  7y  +  29  =  10:?; 

©  ^2  _^  26  =  3xy  +  2y 


®  y  = 

®  y  = 


10^  -  29 

7 
x^  +  26 
3x+2 


[0  ^  29]  -  7 
(2)~-{3x  +  2) 


186 


ALGEBRA, 

® 

10:^- 

7 

29  _ 

^2  +  26 

3:?;  +  2 

® 

23:^2  - 

Qlx 

-  240  :i^  0 

® 

X^b', 

,   X  = 

48 
23 

y  =  3; 

i  y  = 

509 
161 

(D  and  ®  compared 

®  X  7(3:?; -I- 2) . .  . 
®  Ax.  A 


This  method  is  evidently  only  a  slight  variation  of  the 
method  of  substitution.  Sometimes  it  is  useful  to  '  *  com- 
pare "  expressions  more  complex  than  the  simple  x  or  y. 

Model  P. 

©     a;2  +  ^xy  =  144 
©  ^xy  +  36^2  ^  432 


®xy  +  Gtf  =  n 

(D-6 

144 

®    x  +  Qy=t^ 

X 

Q  -^a; 

72 

®  ^2/ 

a;         y 

(T)  and  ®  com- 

(T)    2y:=x 

®  X  a;y  -T-  73 

®  4y^  +  12/  =  144 

®  subst.  in  0 

.-.  y  =:r  ±  3;    :?;  =  ±  6 

201.   This  example  could  also  have  been  done^,  like  num- 
ber 12  above,  by  adding  the  two  equations : 

®  x"^  +  12x1/  +  36y2  =  576  ®  +  ® 

®  X  +  6?/  =  ±  24  (D  Ax.  A. 

(5)    ±  Mx  =  144  ®  subst.  in  ® 

(D  ±  144^  =  432  ®  subst.  in  © 
.\x=  ±  Q;  y  =  ±3 


THE  SECOND  METHOD   OF  ELIMINATION.       187 


EXERCISE  XCV. 


Sot 
1. 

2. 

Ive  hy  comparison : 
y  —  x^  =  2;  y  ~{-  1  =  4:X, 
6y  =  I3x;   y  —  1  =  xK 

3. 

x'  -Z^y-]  y  +  Z  =  x. 

4. 

y  = T ;  y  +  3  =  a;. 

6.  8.T  +  -  =  y  ~  11 ;  7y  =  21  +  65a;. 

21 
6.  23  ~  a;  =  ly\  y 


x-\-b 

7.  :c^  -|-  4  =  %xy]  o;^  4-  9  =  3:?:^. 

8.  a;2  +  36  =  Qxy\  b{x  -  1)  ==  4i/. 

9.  x^  —  bx  =^  {x  -^  3)^;  a;—  3  =  y . 

10.  y{x  —  1)  =  x\  2xy  =  5a;  —  2. 

Quadratics  with  Three  Letters. 

202.  A  system  of  three  equations  with  three  unknown 
letters  can  always  be  solved  if  two  of  them  are  of  the  first 
degree  and  the  other  is  quadratic.  Two  of  the  letters  must 
severally  have  their  values  expressed  in  terms  of  the  third, 
and  these  expressions  must  be  substituted  in  the  quadratic 
equation.  To  this  end  it  is  in  general  necessary  first  to 
eliminate  one  of  the  three  letters  from  the  two  equations  of 
the  first  degree,  and  thus  find  one  expression  for  substitu- 
tion; and  then  to  eliminate  another  letter  from  the  same 
two  equations,  and  get  the  other  expression.  But  it 
sometimes  happens,  as  in  example  1  below,  that  one  or  both 
of  the  expressions  for  substitution  can  be  found  directly 
from  the  given  equations. 


188 


ALGEBRA. 


Model  Q. 

0  2^  +  3?/  -  ^  =  15 
®  ^'  +  2«/  +  3^  =  12 
(D  2i?^''^  —  ^;2  =  48 

®  62;  +  9^  -  3;2  =  45 
®         72;  +  lly  =  57 
(6)  11^  =  57-7:?; 
^  57  -  7:r 

®^  =  -lI- 
(D  4:?;  +  6?/  —  2;^  =  30 
(?)  3.T  +  6^  +  9;^  :=:  36 
@  11;^  —  i?;  =  6 

6  +  a; 


® 

X  3 

© 

+  0 

® 

—  Ix 

® 

-T-     11 

® 

X  3 

® 

X  3 

® 

-® 

(Q)  ^== 

@  2x^  — 


11 

57  -  7:?;    6  +  x 


^--11 


=  48 


subst.  (T)  and  ©  in  (3) 


11  11 

@  2^2x^+7x^-  15x-  342  =:  5808    @  x  121 
@        249:i;2_  I5a^_  6150  =  0        @  -  5808 
@  83:?;2  _  5^.  _  205O  =  0         (U)  ^  3 

®         {83x  +  4:10){x  -  d)  =  0         @  factored 
@  X  =  6;  x  =  —-^^-  from  (16)  by  Ax. 


X            5 

-  w 

y      1      3 

w 

« 

1 

/^ 

u4w5. 


EXERCISE  XCVI. 


3:?;  +  2^  =  12;  3y  +  2;^  =  11;  x'  +  2f  +  3z^  =  25. 
5a;  -  7?/  =  31;  3:^+7;^  =  27;  6xy  -  7yz  +  3:r;2  =  51. 
x^  +  3xy  —  Zxz  =  39;  2:?;  —  3^  =  —  9;  3x  +  4z  —  13. 
3^  +  2^  +  ^  =  23;  :?;  -  3y  +  5^  =  6;  72;2  _  5^;?/  =  100. 


5.  x-y-\-z  =  4:',  3x-  5y  +  2z  =  20;  3x^  -  5yz  =  77. 


THE  SECOND  METHOD   OF  ELIMINATION.      189 

6.  ^  +  y  +  ^-9;  22;+3y  =  13;  ^  +  ?-  =  5. 

7.  x^  -\-y^-\-xz=  24;  ^  +  3y  +  5^  =  22;  2y  -  Zz  =  0. 

8.  X -{-  ij  +  z  =  10',  x^  -\-  y^  +  z^  =  38;  3:?;  +  2^  -  5^  =  4. 

9.  ^  4-  3?/  =  11;  2x  —  z  —  0;  3^/''^  —  y^  =  15. 

10.  a:  +  ^  +  5=2/  +  ;^  +  2=  -^±^  =  15. 


CHAPTER  YIII. 

MULTIPLICATION  OF  FRACTIONS;   HIGHEST 
COMMON   FACTOR. 

Fractions  and  Ratios. 

203.  Whenever  the  operation  of  division  in  Algebra  can- 
not be  carried  out,  or  whenever  for  any  other  reason  it  is 
desirable  to  indicate  division,  the  dividend  may  be  written 
as  a  numerator,  and  the  divisor  as  a  denominator,  and  the 
whole  result  is  called  a  fraction, — or,  sometimes,  a  ratio. 

204.  It  must  be  remembered  that  there  are  two  kinds  of 
division.     For  example  : 

The  expression  -\^-  may  mean, — 

*^If  I  divide  30  separate  objects  into  5  equal  parts,  how 
many  separate  objects  will  be  in  each  part  ?  ^' 

Or,  otherwise,  the  expression  -\Q-  may  mean, — 

'*How  many  equal  portions,  consisting  each  of  five  sepa- 
rate objects,  may  be  gotten  out  of  a  collection  of  30  sepa- 
rate objects  ?  ^' 

205.  In  the  first  case  the  divisor  is  an  abstract  number 
and  the  quotient  is  of  the  same  denomination  as  the  divi- 
dend. •  The  indication  of  such  division  is  in  Algebra  prop- 
erly called  a  fraction. 

206.  In  the  second  case  the  divisor  is  of  the  same  denom- 
ination as  the  dividend,  and  the  quotient  is  an  abstract 

190 


MULTIPLICATION  OF  FRACTIONS.  191 

number.     The  indication  of   such  division  is  in  Algebra 
properly  called  a  ratio. 

207.  In  multiplication  the  multiplier  is  always  an  ab- 
stract number,  and  the  product  is  always  of  the  same 
denomination  as  the  multiplicand.  In  reversing  the  pro- 
cess of  multiplying  (that  is  to  say,  in  dividing),  when  we 
get  the  multiplicand  for  our  quotient  we  have  a  fraction 
of  the  product,  and  when  we  get  the  multiplier,  we  have 
the  ratio  of  the  product  to  the  multiplicand. 

208.  Again,  whenever  we  ask  how  long,  or  how  large,  or 
how  heavy  anything  is,  the  numerical  answer  to  our  ques- 
tion is  always  a  ratio, — the  ratio  of  the  quantity  we  are 
asking  about  to  the  yard  or  the  mile,  to  the  acre  or  the 
square  inch,  to  the  ounce,  the  pound,  or  the  ton. 

209.  The  student  has  probably  seen  by  this  time  that 
Algebra  does  not  concern  itself  with  denominations,  so  that 
we  cannot  tell  in  any  example  of  division  whether  that 
particular  division  results  in  a  fraction  or  in  a  ratio — unless 
we  happen  to  know  in  advance  just  what  our  letters  repre- 
sent. But  it  is  not  necessary  to  know.  All  the  laws  of 
transformation  are  the  same  for  fractions  as  for  ratios. 
Where  it  is  very  necessary  to  distinguish  a  ratio  it  may  be 

written  a  :  b  instead  of  - ;  but  generally  in  Algebra  such  a 

distinction  is  not  worth  while. 

210.  The  expression  a  :  Z>  (or  -J  is  read  ^^the  ratio  of 

a  to  Z>^';  ^  is  the  antecedent,  the  dividend,  or  the  numerator, 
and  b  is  the  consequent,  the  divisor,  or  the  denominator,  of 

the  ratio.     The  form  -  is  generally  read   '  ^  a  over  b  '^  or, 

more  fully,  ^' a  divided  by  Z>."     The  term  FRACTioisr,  in" 

THIS    CHAPTER     Al^D     THE     SUCCEEDING     OKE,     WILL    BE 
UNDERSTOOD   TO   INCLUDE   RATIOS. 


192  ALGEBRA. 

211.  The  ratio  of  any  number  to  one  is  the  number 
itself  ;  consequently  any  number  can  be  regarded  as  having 
a  denominator  one. 

EXERCISE  XCVII. 

1.  The  ratio  of  two  numbers  is  |,  and  the  smaller  is  3; 
what  is  the  larger  ? 

2.  The  ratio  of  two  numbers  is  9,  and  the  smaller  is  3; 
what  is  the  larger  ? 

3.  The  sum  of  two  numbers  is  20,  and  their  ratio  is  3; 
what  are  they  ? 

4.  The  sum  of  two  numbers  is  100,  and  their  ratio  is 
the  same  as  the  ratio  of  one  of  them  to  100.  Find  the 
value  of  the  numbers  within  one- tenth. 

5.  One  number  is  12  greater  than  the  square  of  another, 
and  their  ratio  is  7.     Find  the  numbers. 


THE   LAWS   OF   FRACTIONS. 

212.  Theorems  which  are  of  use  in  this  chapter  and  the 
succeeding  one  are  proved  by  the  use  of  the  three  funda- 
mental laws  of  Algebra.* 

213.  Just  as  the  sign  —  means  the  reverse  of  addition, 
addition  undone,  so  the  sign  -^  means  the  reverse  of 
multiplication,  multiplication  undone.  The  sign  —  may 
be  read  ^^  the  negative  of,^'  and  the  sign  ~  may  be  read 
''  the  reciprocal  of." 

214.  The  law  of  association  for  +  and  —  signs  implies 
such  identities  as  these: 

a+l-c-\-d-e^{a+V)-{c-d^e)=a+{l-c)-\-{d-'e) 

*  See  §  86. 


MXILTIPLICATION  OF  FHAGTIONS.  193 

In  the  same  way  the  law  of  association  for  -~  and  X  im- 
plies : 

axi-^cxd-^e={axi)-i-{c-^dxe)  =  ax{l>-^c)x{d-^e) 

In  fact,  THE  ASSOCIATIYE  AND  DISTRIBUTIVE  LAWS 
ARE,  SO  FAR  AS  FORM  GOES,  THE  SAME  FOR  X  AND  -f- 
AS   THEY    ARE    FOR    +    ^^^    —  • 

215.  Theorem  I.  Any  fraction  may  have  its  numerator 
and  denominator  multiplied  or  divided  hy  the  same  numher 
without  altering  its  value. 

Let  —  be  any  fraction  and  let  m  be  any  number.     Then 

am  ^  ,,  . 

.: —  =  am  ~-  am  =  a  X  m  -^  (o  X  m) 
om  ^  ^ 

=  a  X  m-^Z>-^m[the  distributive  law] 
=  a  -^  b  X  m  -^  m  [the  commutative  law] 

.    7.    -^ 
=  a  -^  0  —  Y 

0 


hn       h 

216.  Theorem  II.  Fractions  are  multiplied  by  multi- 
plying their  numerators  together  for  a  new  numerator 
and  their  denominators  for  a  new  denominator. 

a  c 

Let  7-  and  -^  be  any  two  fractions. 

a        c 

-rX-r  =  «-^^  X  G  -^  d^  a  X  c  -^  h  -^  d 

0       d 

=  aXc-^{l)Xd)~ 


U 


a        c  _  ac 
I  ^  d^M 


194  ALGEBRA, 

217.  Theorem  III.     The  rule  for  dividing  by  a  fraction 
is :  Invert  the  divisor  and  then  multiply. 

Let  X  be  any  number  and  —  any  fraction. 

h 

=.  X  X  - 

a 

a  i 

.*.  X  -^  T-  -  X  X  - 

0  a 

Another  way  of  stating  the  rule  is  this:  To  divide  by 
any  number,  multiply  by  its  reciprocal. 


Reduction  of  Multiplications. 

218.   The  first  step  in  every  example  like  the  following 
is  to  factor  numerator  and  denominator  in  every  fraction : 


Model  A. 

fx''  - 
'\x^- 


5a;  -  66  x''  -  2x  fx''  -  121      x"^  -\- bx  -  66\ 

a.2  _  5a.  -j_  6  '^  ic^  +  6a;  -  72  ■   V  a;^  -  9'  '^  (c'  +  9a;  +  isj 

^  {X  -^l)(aH-6)  ^^       x{x  -  2)        ^  (a;  +  ll)(a;-ll)      (a;+ ll)(a;-6) 

.  X  / 


^  a!2  +  6a; 

- 

72  • 

x(x  — 

2) 

(a; +  12)0 

X  — 

6)- 

x(x  — 

^ 

^  X 

-  {x  -2)(a;  -  3)^(a;-f  12)(a;-6)  *   (a; -f  3)(a;  -  3)  ^  (a;  +  3)(a;  +  6) 

^  (a;-ll)(a;+6)         a;(a;  -  2)  (a;  +  3)(a;-3)    ^  (a;  +  ll)(a;-6) 

-  (x  -2)(a;  -  3)^(a;+12)(a;-6)^  (a;4- ll)(a;  -  11) '^  (a;  +  3Xa;-f6) 

According  to  Theorem  II  all  the  factors  in  the  numera- 
tors, as  the  several  fractions  now  stand,  are  factors  of  the 
numerator  of  the  product  which  is  the  answer;  and  the 
same  for  the  denominators. 

According  to  Theorem  I  we  may  divide  the  numerator 
and  denominator  of  the  product  by  the  same  number 
without  altering   its  value;    that  is,  we   may   strike   out 


MULTIPLICATION  OF  FRACTIONS,  195 

any  factor  that  is  the  same  in  both.     The  expression  in 
Model  A  then  becomes 

(a;-ll)(a;+6)         x{x  -  2)         {x  +  Z){x  -  3)      (x-\-n){x-Q)  _     x 
{x-2){x-d)  '  {x+12){x-Q)  '  (a;+ll)(lc-ll)  '  (ic+3)(a;+6)  ""  ic+12 

Ans. 

EXERCISE  XCVIII. 

Perform  the  operations  indicated  : 

3x        7y    .  4:X^  a  —  b         n^  —  P 

^'  4^  ^12x'^  9f'  ^'  a^  +  ad  ^  a^  -  ab' 

x^  +  X  —  2   ^         x^  -{-  2x 

""^^  -  7x  ^  x^  -  Idx  -f  43* 
,  a;^  +  3a:  +  2  x^  -  7^:  +  12 
'  x^  —  6x  -{-  Q  x^  -\-  X 

^  4:X  +  3      x^  —  9x  +  20      x'^  —  "Tx 


3. 


x^  —  bx  -{-  4:     x^  ~  10:r  +  21     x^  —  5x ' 
1  1 


a'  -  Via  +  30  *  a-  15* 
tt'^  —  2ay  -\-  y^  —  W-  ^  a  —  y  -\-'b 
a^  +  2ay  -\-  y^  —  V  '  a  +  y  —  b' 
a'  +  a^  -  SaW  +  19^^?^  -  lb¥ 
d^  +  'dab  -  bl)^ 

X 


X 

9.  :j — .    [Multiply  numerator  and  denominator  by  x^,  ] 


X 


3ax       a^  —  x^       f  a^  -^  ax      a  —■  x 

iby       c^  —  x^  '    \bc  -^  bx      c  —  x. 

^^-   l,a2  +  a^  +  ^^V    *  a  +  Z^^  a^  +  F 

1  +  2/      ^  +  ^^ 
^^-      4a2_i      -2^+1^  a* 


196 


ALOEBBA, 


14. 


15. 


16. 


X^  —  X  —  W  X^  —  X 

X 


17. 


19. 


20. 


■  2 


x  +  1 

x'^  +  'Zx-^'^  x^-\-  bx 


X 


^01? 


X 


X''  —  25 
4:X^  -j-  :r  —  14 
6x1/  —  14?/ 

x^  -]-  X  —  2 

x^  ^x-  20  "^  \^a;^  -  2^  -  15 

2:2  _  isx  +  80       2;2  ~  15^;  +  56 


X  —  2        3x^  —  X—  14: 


X^    —    4:    ^^    4:X 

'x^  +  3^:  +  2 


X 


7^      2x^  +  4.x 
X  -{-  d       x^  +  6x+4:' 


x^ 


•)■ 


X 


1 


18. 


6x^  - 


^bx-bO 
^.  -  2a^ 


x^-6x-7      *  .T  +  5* 
X  —  a        3ax  +  2d^ 


ax  —  a'^ 
2:^  —  64 


9x^  —  4:d^       2x  -\-  a 

,2\ 


'x       x^  —  dxy  +  9y^\ 


{x  +  3yy 
x^j^l2x-Q4: 


x^  _[_  24tx  +  128        x^ 


64 

{x  -  sy 

_X^-\-      4:X    +    16 


x^-  U       J  * 


One  Term  Not  Easily  Factorable. 
219.   Sometimes  in  reducing  fractions  to  lowest   terms 
only  one  of  the  terms  of  the  fraction  can  be  factored  by 
inspection.     Then  it  is  best  to  try  the  separate  factors  as 
divisors  of  the  unfactored  term. 

x^  +  2^:  —  3 


Model  B. — Keduce  to  lowest  terms 


3^3  _|.  7^  __  10' 


x^+2x  -  d   _{x  +  3){x  -  1) 


3^3  _^  7^  _  10  -  3x^  +  7a:  -  10  • 

Evidently  x  +  3  cannot  divide  the  denominator  exactly 
because  3  is  not  an  exact  divisor  of 
the   straight  product  10.     Dividing 
by  ic  —  1  we  get  for  the  answer 

X  +  3 


3x^  +  3x  +  10' 


X  -  1 

3^2  _^    ^^   _^ 

10 

3^3  _!_  7^  _ 
3x^  -  3x^ 

10 

3x^  +  Ix 
3x'^  -  3x 

lOx  -  10 


MULTIPLICATION  OF  FBAGTIONS,  197 

EXERCISE  XCIX. 

Reduce  to  loivest  terms  : 

Sd^  -  6ab  x^  -  6x  +  4. 

2d^b  -  4:ab^'  ^'  4:X^  +  9x  —  13* 

a{2b'^  -  2ba)  2x^  -  1032;  +  35 

b{ib^a  -  9a*)*  *''     x^  -  Ux  +  42  * 

3^4  _|_  Q^^y  _^  Q^2y2  3a;3  -  78^;  +  15 

^4  _j_  ^3^^  _  2x'^i/'^  '                 ®*   2a:^  —  13a;  +  15* 

x^  +  xy  —  2y^  x^  —  3x  —  10 


x^  -  f       '  Ix^  -  232;  +  10* 

27a  +  «'  62;2  +  c?;  -  12 

! in       1 . 

18a  -  6a^  +  2a^  12^:3  __  23^;  +  6 


Test  for  Simple  Factors. 

220.  It  is  sometimes  easy  to  decide  whether  a  very  simple 
factor  is  contained  without  remainder  in  an  algebraic  ex- 
pression without  performing  the  operation  of  division. 
The  test  depends  upon  the  axiom  that  if  one  factor  of  an  ex- 
pression is  equal  to  zero,  then  the  whole  expression  must  be. 

Model  C. — To  decide  whether  a;  —  1  is  a  factor  of 
^3  _|_  5^2  _|_  ^  _  10  it  is  only  necessary  to  suppose  that 
a;  =  1 ;  in  that  case  a;  —  1  =  0  and  x^  -\-  bx^  -\-  x  —  10 
=  7  —  10=—  3;  hence  c^:;  —  1  is  not  a  factor,  because 
when  X  —  1  =  0  the  expression  x^  +  bx^  -\-  x  —  \0  \^  not. 

In  the  same  way  x  -\-  A  is  not  a  factor,  because,  if 
a:=  —1,  x^-\-bx^-\-x~lQ  becomes  —1  +  5—1  —  10=:  —7; 
and  a;  —  2  is  not  a  factor,  because,  if  a;  =  2,  x^-\-bx^-^x—10 
becomes  8  +  20  +  2  —  10  =  20 ;  but  a;  +  2  is  a  factor, 
because,  if  x  =  —  2,  x^  -\-  bx'^  -\-  x  --  10  becomes 
_  8  +  20  -  2  -  10  =  0. 


198  ALGEBRA, 


EXERCISE    C. 


In  the  same  way  decide  luhether  either  or  both  of  the 
binomials  given  loith  each  of  the  folloioing  expressions  are 
factors  of  that  expression  or  not : 


Possible  Factors. 

1. 

^3  _|..  2x^  -  ^x  -  6 

:^  + 

1 

X  -2 

2. 

x^  -  2x^  -  5x  +  Q 

x  + 

3 

x  +  2 

3. 

x'  -  Ux  -  12 

x  + 

1 

X  -  1 

4. 

x^  -  7x  +  6 

X  — 

2 

X  -    I 

5. 

^3  _    3^2  ^  4 

X  — 

2 

x  +  2 

6. 

x^  -  19:?;  +  30 

X   — 

2 

X  -  3 

7. 

^3  _  5^2  _  2x  -f  24 

X   — 

2 

X  +  3 

8. 

x^  -  Ux^  -f  Ux  ~  24 

x  + 

1 

X  -  2 

9. 

x'+  x^  -  10303:?;^  +  ^  -  10302             x  -j- 

1 

X  -  1 

10. 

3^-4  _  7^2  _  20 

x  + 

2 

X  -2 

11. 

X''  —  a" 

x  + 

« 

X  —  a 

12. 

x""  -\-  a" 

x  + 

a 

X  —  a 

221.  Theoretically  the  same  test  would  serve  to  decide 
whether  3a;  —  2  was  a  factor  of  3x*  +  13:?:'^  —  11:?;  —  18; 
but  practically  it  is  easier,  in  this  case,  to  perform  the 
operation  of  long  division  than  to  substitute  in  the  given 
expression  x  =  ^.  The  pupil  must  use  his  judgment  as  to 
which  test  he  shall  apply  in  each  example. 

EXERCISE  CI. 

Fi7id  the  IT,  C.  F,  in  each  of  the  following  examples  : 

1.  2x^-3x^-5x-i-6;  x^-x-2,     3.  :?;2-3a;  +  2;  7:?;^4-13a;2-20. 

2.  5:c3-21a;2+16;  a;''^-:2:-12.     4.   3x^-{-2x' -^7x-M]x:^-4, 

6.  3x'  +  7x-Q2;  x^-\-29x-62. 


HIGHEST  COMMON  FACTOR.  199 

Neither  Term  Easily  Factorable. 

222.*  In  the  following  example  it  is  extremely  difficult 
to  factor  either  the  numerator  or  the  denominator  by  in- 
spection. 

,,  .  ,  Tx      T.  1        X    .         XX  182;3  +  IZx  -  14 

Model  D. — Reduce  to  lowest  terms     ^,  »   ,   ^  >. —, 

24:X^  +  2:z;^  ~  8 

We  must  try,  then,  in  some  other  way  to  find  a  common 
divisor  for  these  two  expressions.  The  way  we  shall  adopt 
is  to  change  these  expressions  into  others  which  have  the 
same  common  divisors  and  at  the  same  time  are  easier  to 
factor. 

Let  us  take  as  an  abbreviation  for  the  numerator  the 
letter  P,  and  for  the  denominator  Q. 

Q  has  a  factor  2,  which  is  isroT  a  factor  of  P  and  is  there- 
fore not  a  common  factor.     Divide  Q  by  2,  and  we  get 

12a;3  +  x^  -  4:; 

multiply  this  by  3,  and  we  get 

36a;3  +  3a;2  -  12. 

Since  we  have  not  introduced  or  taken  out  common 
factors,  any  common  divisor  of  F  and  Q  is  also  a  common 
divisor  of  F  and  36x^  +  dx^  -  12. 

In  the  same  way  2  X  P  =  36a;3  -f  26a;  —  28. 

Any  number  that  will  divide  each  of  two  quantities  will 
also  divide  their  sum  or  their  difference.  [This  is  merely 
one  form  of  the  distkibutive  law.  We  may  illustrate  it, 
but  we  cannot  prove  it,  because  it  is  one  of  those  funda- 
mental principles  that  have  to  be  taken  for  a  basis  of  all 
proofs.] 

*  See  also  §§  229,  230. 


200 


ALOEBRA. 


So  any  number  that  is  a  divisor  of  P  and  Q  is  also  a 
divisor  of  362;^  +  32;^  —  12  and  of  362;^  +  262;  -  28;  and 
is  therefore  a  divisor  of  their  differekce,  that  is,  of 
32;2  _  262;  +  16. 

The  advantage  of  this  result  lies  in  the  fact  that  we  have 
ELiMi^STATED,  SO  to  speak,  the  x^  terms  from  our  two  ex- 
pressions, and  so  obtained  a  quadratic  expression,  which 
contains  among  its  factors  every  common  factor  of  P  and 
Q.     Factoring, 

32;2  -  262;  +  16  :E  (32;  -  2)(2;  -  8). 


We  see  that  2;  —  8  cannot  be  a  common  factor,  because 
8  will  not  divide  14,  one  of  the  straight  products.  Divid- 
ing numerator  and  de- 
nominator by  32;  —  2,  we 
get  for  the  answer  to  the 
example  : 

62;'-^  +  42;  +  y 
82;'-^  +  62;  +  4 


32; -2 

32;- 2 

62;2  +  42;  +  7 

82;^  +  62;  +  4 

182;3  +  132;  - 
182;^  -  122;2 

14 

242;^  +  22;2  -  8 
242;3  -  162;'^ 

122;2  +  132;  - 
122;2  -  82; 

14 

182;^  -  8 
182;'^  -  122; 

2I2;  -  14 
2I2;  -  14 

122;  -  8 
122;-  8 

0 

0 

EXERCISE  CII. 


Reduce  to  lowest  terms  : 
32;^  -  112;^  +  18 
^*  22;3  -  232;  +  15  ' 

32;^  +  232;^  -  50 
^'  92;3  -  192;  +  10  • 

122;^  -  3I2;  +  6 
^-  82;^  -  282;  +  15* 


212;^  -  92:y^  +  8 
492;^  -  I82;  -  4  ' 

252;^  -  602;^  +  49 
252;^  -  642;2  +  21* 


HIGHEST  COMMON  FACTOR.  201 


More  Difficult  Examples. 

223.  When  we  have  for  terms  of  our  fraction  expres- 
sions of  degree  higher  than  3,  this  process  will  generally 
have  to  be  carried  a  little  further. 

Model  E.-Reduce  ""^T^l^^l- 

Here  we  destroy  the  highest  terms  as  before.  For 
convenience  we  may  number  the  expressions  and  make  a 
memorandum  of  how  we  get  them,  just  as  we  do  with  equa- 
tions : 

®2x^  -  Qx-20  ©  X    2 

®  5a;3  -  6z  -  28  (D  -  © 

This  (4)  is  an  expression  of  the  third  degree,  which  we 
cannot  easily  factor.  We  know,  however,  that  it  contains 
all  the  factors  common  to  0  and  @.  We  can  obtain  an- 
other such  expression  by  destroying  (^^eliminating")  the 
lowest  terms. 

d)  4^;*  —  12^  -  40  ®  X   4 

(D  lOx"^  -  26x^  +  40  ©  X   5 

©  14:x^  -  25:^3  -  12^  ®  +  ® 

This  contains  a  factor  x,  and  as  no  factor  x  occurs  in 
either  of  the  expressions  we  are  investigating,  we  shall  not 
affect  our  result  if  we  cast  it  out. 

®  Ux^  -  25r?;2  -12  (2)^x. 

By  the  same  reasoning  as  before,  ®  contains  all  the  fac- 
tors that  are  common  to  ®  and  ©.  Therefore  the  H.C.F. 
of  ®  and  ®,  if  we  could  find  it,  would  contain  the  H.C.F. 


202  ALGEBRA. 

of  0  and  ®.*    We  now  apply  ourselves  to  a  new  problem, 
—to  find  the  H.  C.  F.  of  ©  and  ®. 
®  bx^  -  6:r  -  28 
®  I4.x^  -  25:^2  __  12 
®  70a;3  -  842;  -  392  ®  X  14 

@  lOx^  ~  1252;2  -  60  ®  X     5 

(0)  1252;2  -  84^:  -  332  ®  -  (TO) 

This  expression  (Tj),  though  only  a  quadratic,  is  still  diffi- 
cult to  factor.  We  are  aided  by  the  fact  that  0  and  @, 
the  given  numbers,  have  for  straight  products  x"^  and  2ic^ 
—  10  and  8,  and  consequently  if  there  is  a  common  factor 
which  is  also  a  factor  of  (?]),  that  factor  may  be  a;  +  ^ 
or  ic  —  2.  By  trial  we  find  the  factors  of  @  to  be 
(125^:  +  166) (a;  —  2).  If  there  is  a  common  factor,  then, 
of  0  and  ©,  it  must  be  a;  —  2.     Dividing  : 

^j-2 a; -2 

x^ -\- "^x^ -\- ^x -\- b     2x^—x^  —  2x~4i 
x*-3x  —  10  2x^  -  5^;^  +  8 

^'r^^' ^^'  7  ^"^^ ,      ^3_|_2^2_|_4^_^5 

22;3  _  32;  -  10  -  2;^  +  8  ^?^s.  ,-f — rS"^- 

22;^  -  42;'^ -  x^  +  2x^  ix^x-Ax-^ 

42;2  -  32;  -  10  -  22;^  +  8 

42;2  -82;  -  22;2  +  42; 


52;  ~  10  -  42;  +  8 

EXERCISE   cm. 

Reduce  to  loiuest  terms  : 

x^  -f  32;^  -  272;  +  14  1  +  22;^  +  x?  +  2x^ 

1-         ^4_i5^_^x4       •          ^'  1  +  3^2_|.2^3^3^4- 

42;*  +  ll2;2  +  25  62;^  4-  2;^  —  52;  —  2 

^'  42;*  _  92:2  _j_  30^  _  25*        ^'  62;^  +  6x^  -  32;  -  2' 
9;^4  _  6^3  _  343 
5. 


92;*  -  492;2  +  62;  +  14 
*  So  far  as  we  have  yet  proved,  it  miglit  contain  other  factors, 
further. 


HIGHEST  COMMON  FACTOR.  203 

Theory  of  H.  C.  F.  by  Elimination. 

224.  If  P  and  Q  be  the  numerator  and  denominator, 
respectively,  of  a  fraction  which  we  are  to  reduce  to  lowest 
terms,  the  process  used  above  may  be  carried  out  in  sym- 
bols, and  some  general  conclusions  can  be  drawn  from  the 
formulae  thus  obtained. 

To  destroy  (or  eliminate)  the  terms  of  highest  degree  we 
multiply  P  and  Q^  respectively,  by  some  suitable  multipliers, 
say  a  and  d,  and  subtract.  The  result,  which  we  may  call 
X,  is  an  expression  of  lower  degree  than  P  and  Q, 

(\)X=aP-bQ. 

Then  destroying  the  terms  of  lowest  degree,  and  letting 
h  and  Jc  stand  for  our  two  multipliers : 

Now  it  is  evidelit  from  0  and  (2)  that  any  common  factor 
of  P  and  ^  is  a  factor  of  X,  and  also  a  factor  of  F,  that  is, 
a  common  factor  of  X  and  Y,  Let  us  take  these  two 
equations  and  find  an  expression  for  P  and  Q,  to  see  if 
some  other  conclusion  can  be  drawn. 

(D    xY  =  hP  -  JcQ 

®    hX  =  ahP  -  bJiQ 

(b)  axY  =  aliP  —  akQ 

(6)  hX  -  axY  =  akQ  -  hhQ 

^hX-axY       ^ 

®    ah-hh     -^ 
Here  we  have  found  an  expression  for  Q  by  eliminating 
P,  and  from  that  value  of  Q  it  is  evident  that  any  common 
factor  of  X  and  Y  is  also  a  factor  of  Q.     In  the  same  way 
it  would  be  found  that 

^^  -     ak-  bh    ' 


®  X  a; 

(J)Xh 

®  X  a 

®-® 

(D  H-  (a/fc  - 

-  bh) 

204  ALGEBRA. 

and  it  is  established  that  any  common  factor  of  X  and  Y  is 

also  a  factor  of  P. 

Therefore,  since  X  and  Y  have  for  common  factors  all 

the  common  factors  of  P  and  Q,  and  no  others,  the  H.  C.  F. 

of  X  and  Y  is  the  same  as  that  of  P  and    Q\  and  the 

elimination  may  be  continued,  taking  each  pair  of  results 

as  a  new  problem,  until  the  H.  C.  F.  appears  in  two  identical 

expressions. 
Model  F. 

©  Let  P  =  4^;*  +  26a;3  +  IW  -  2x  -  U 

@  Q  =  3x^  +  20^-3  +  32x^  -Sx-  32 

(D  12.^4  +  78i?;^  +  123^;'^  -  6a;  -  72  ®  X  3 

®  12a;4  +  80a;3  +  128:?;'^  -  32x  -  128         ©  X  4 

®  2:^3  +  5^"^  -  26a;  -  56  X=:4(>-3P 

®  16a;4  +  104a;3  +  164^;^  -  8a;  -  96        ©  X  4 

©    9x^  +  60x^  +  96x^  _  24a;  -  96  @  X  3 

®    7x^  +  Ux^  +  68a;2  +  16a;  ®  -  © 

®    7a;3  +  44a;^  +  68a;  +  16      ®~  x   Y=  ^ ^ 

a; 

Now  since  the  H.  0.  F.  of  Xand  F  is  the  same  as  the 
H.  C.  F.  of  P  and  Q,  we  may  start  on  a  new  problem, 
namely,  to  find  the  H.  C.  F.  of  20;^  +  bx^  -  26a;  -  56  and 
7a;^  +  44a;^  +  ^^^  +  1^^  which  we  may  call  P  and  Q 
respectively. 

@  Let  P=2x^  +  5x^  -  26a;  -  56 

(0  Q  =  7x^  +  Ux^  +  68a;  +  16 

(jD  i4a;3  _(.  35^2  _  182^  _  392     @  x  7 

(@)  14a;3  +  88a:2  _f_  2.36a;  +  32       (ij)  X  2 

@  53a;2  _|_  313^  ^  424  x=  2§  -  7P 

@  4a;3  +  10a;^- 52a;- 112        @X2 

@  i^x^  +  308a;'^  +  476a;  +112   ([j)  X  7 

@  53a;3  +  318a;'^  +  424a;  ®  +  ® 

®  53a;2  +  318a;  +  424  -p^2P+7Q 


HIGHEST  COMMON  FACTOR,  205 

Now  since  X  and  Y  are  the  same  expression,  their 
H.  C.  F.  is  that  expression,  and  that  is  the  H.  0.  F.  of  P 
and  Q,  —  and  that  is  the  H.  G.  F.  sought. 

EXERCISE   CIV. 

Reduce  ivherever  possible  : 

2x^  -  lla;''^  -  9  x^  —  lOo;^  +  9 

^'  4:X^  +  lla;^  +  81*  ^'  x^  +  Wx^  +  20:?;2  _  iq^  _  21* 

9^4  __  2x^  A-x^  ^^x  —  6  x^+  5x^  —  x^  —  5x 


8 


4^4  _  2x^  -f  3^  —  9     '       ^'     x^  +  ^x^  —  x—'d 
;3  _|_  2^3  +  a;  +  3  ^5  _  16^  _  32 


■  x 

X 


*•  4^3  -  18:?;2  _|_  19^  _  3'  9.   32^3  _  24^2  _  g^  +  3* 

32;^  -3x3  -"Ix^  -x-\  x^  -x^-2x  +  'il 

^'    Qx^-3x^-x'-x-l'      ^^'       U^-x-l     * 

Another  Application  of  H.  C.  F. 

225.  We  have  seen  that  an  algebraic  equation  with  one 
unknown  letter  can  have  no  answers  but  those  which  make 
the  several  factors  equal  to  zero.  If  there  are  two  such 
equations,  then,  which  can  be  satisfied  by  the  same  value 
of  X,  it  follows  that  the  two  equations  must  have  a  common 
factor. 

Model  G. — There  is  a  number  which  will  satisfy  each  of 
the  two  equations 

16x*  -  8x  +  3  =  0 
64^3  =  8 

To  find  what  that  answer  is,  we  first  find  the  H.  C.  F.  of 
the  two  expressions  16x^  —  8a;  +  3  and  64^3  —  8. 

©  16x^  -  8x  +  3  =  0 

©  64^3  -8  =  0 

(D  8x3  -  1  (D  -^  8 


206  ALGEBRA, 

®  24:X^  -  3  (D  X  3 

®  16x^  +  242)3  _  g^  ®  +  © 

'^®2x^  +  3x^  -  1  ®  -^8x 

(T)  16x^  -  2x  ®  X2x 

®  6^  -  3  ®  -  ®  ° 

*  ®  2:?;  -  1  ®  -^  3 

If  there  is  an  H.  C.  F.  it  must  be  2r?;  —  1 ;  we  find  on  trial 
that  this  is  contained  in  2x^  -\-  Sx^  —  1  exactly  x^  -\-  2x  -\-  1 
times;  hence  2:^  —  1  is  a  factor  of  both  equations  ®  and 
©.  That  is,  each  equation  will  be  satisfied  if  2:x;  —  1  =  0; 
hence  x  =  ^is  a,  root  of  each  equation, 

EXERCISE  CV. 

In  each  of  the  folloiviiig  examples  there  is  one  numher 
that  will  satisfy  both  equations.     Find  that  numher, 

1.  x^  -  nx"  +  \^x  -  12  =  0;  3^:2  -  14a;  +  16  =  0. 

2.  x^  -M^  -    9;r  +  27  =  0 ;  ^2  _  2:^;  _  3  :=  0. 

3.  2x?-\-^x^  -^x-    9  =  0;  32;3  +  "^x^  -  \\x  -  15  =  0. 

4.  x^  +  19a;2-22^  -  40  =  0;  x^  ^  IW  -    6^  -  40  =  0. 

5.  4^3  -  2\x  +  10  =  0;  2x^  +  9a;2  _  25  =  0. 

6.  x^  -  \W  +  18:r  -  8  =  0;  ^x^  -  22:r  +  18  =  0. 

7.  a;*  -  2x^  -  x^  -  ix -^\2  =  ^',  2x^  -Zx^  -x-2^  0. 

8.  x^'-lx^^XW^Zx-X^^^)  42;3-21.t2+26:?;+3  =  0. 

9.  x^  ~  ^x^  -  ^x^  +  36:?;  -  27  =  0;  x^-Zx^-'^x^^'  =  0. 
10.  a;^+13a;3+332;2+31a;+10=0;  ^x^-\-'^^x^-\-^^x^Z\=^^. 


HIGHE8T  COMMON  FACTOR.  207 

H.  C.  F.  of  Three  Expressions. 
226.  Model  H.— Find  the  H.  C.  F.  of 

Sx^  +  dx^  -  64:X^  +  9;     9x^  -  64:X^  +  dx  +  8;  and 

x^  J^  x^  -  8x^  -  5x  +  3. 

QSx^ +  dx^-  Ux^ +  9 

(2)x^  +  x^  -  8x^  -  bx  +  3 

(D  8x^  +  8x^  -  64:X^  -40^  +  24  ®  X  8 

®  5x^  -  40a;  +  15  ®  -  (D 

*®x^-8x  +  3  ® -^  5 

®  3x^  +  dx^  -  24:X^  -  15:^;  4-  9  ®  X  3 

®  5x^  -  4:0x^  +  15:?;  ®  -  ® 

*®x^  -  8x  +  3  (T)  -^  6x 

It  is  now  clear  that  x^  —  Sx  -\-  S  is  the  H.  0.  F.  of  two 
of  the  given  expressions;  it  contains,  then,  all  factors  com- 
mon to  those  two.  The  H.  C.  F.  of  this  result  and  the 
remaining  one  of  the  three  given  expressions  will  contain 
all  the  factors  common  to  the  three;  so  we  seek  the 
H.  C.  F.  of 

x^  -  8x  +  3     and     9a;*  -  64:X^  +  dx  +  8. 

Qx^  -  8x  +  3 
®  9x^  -  64:X^  +  3x  +  8 

@  9x^  -  72.^'^  +  27x  ®  X  9a; 

®  8x^  -  24a;  +  8  ®  -  ® 

*®a;2-3a;  +  l  ®-^8 

The  quadratic  expression  a;^  —  3a;  +  1  is  prime;  hence 
if  there  is  an  H.  C.  F.,  this  must  be  it.  Dividing,  we  find 
that 

Hence  a;^  —  3a;  +  1  is  the  H.  0.  F.  required. 


208  ALGEBRA. 

EXERCISE    CVI. 

Find  the  H,  C.  F,  of: 

1.  a^-x-%)  ^x^  -x^  —  lSx  +  ^]  2^*  +  3ic3  +  .r2-9:z;-9. 

2.  2x^  -  ^x^  +  13a;  -  6;  4.x^  -  20^^'-^  +  31:^;  -  15; 

2a;3  -  ll2;2  4-  19.-?;  ~  10. 

3.  9^*  -  lOx^  +  1;  21^;*  +  10:^:3  _  <^q^2  _  iq^^  -  1; 

21^4  _  4^3  _  222;2  J[.  4:X  +  \. 

4.  6a;  -  ll^''^  +  6a;3  -  1;  19:?;2  +  l  _  4,x{^x^  +  2); 

3:^;(3  +  ^x^)  -  (1  +  262^^). 
6.  a;3  -  7a;  --  6;  2x^  -  Ix^  +  9;  "^x^  -  Sic'-^  -  9a;  +  18. 

H.  C.  F.  by  Long  Division. 

227.  In  Arithmetic  the  H.  C.  F.  of  any  two  numbers  is 
always  found  by  dividing  the  larger  by  the  smaller,  the 
divisor  by  the  remainder,  and  so  on,  continuing  the  opera- 
tion until  there  is  no  remainder. 

Model  L— To  find  the  H.  0.  F.  of  62651  and  18377 

18377)62651(3  =  q^ 
55131 
r,  =  7520)18377(2  =  q, 
15040 
r^  =  3337)7520(2  =  q^ 
6674 
7-3  =  846)3337(3  =  q^ 
2538 
r^  =:  799)846(1  =  q^ 
799 
r,  =  47)799(17  =  q, 
47 
329 
329 
0 


HI0HE8T  COMMON  FACTOR, 


209 


A  much  more  compact  arrangement  of  the  work  is  here 
shown.  [The  successive  quotients  are  lettered  q^^q^,  ^39  •  •  • 
and  the  successive  remainders  r^,  r^,  r^,  •  •  •] 

1  =  ?. 


?.  = 

2 

3 

17 

b 

=  18377 
15040 

62651  =  a 
55131 

?.  = 

r. 

=  3337 
2538 

7520  =  r, 
6674 

?.  = 

r. 

=   799 

47 

846  =  r, 
799 

r. 

=   329 
329 

47  =  r. 

0 

228.  The  theory  of  this  method  is  substantially  the 
same  as  that  of  the  method  by  elimination  of  highest  and 
lowest  terms.  If  a  and  h  are  the  dividend  and  divisor,  and 
q  and  r  the  quotient  and  remainder,  respectively,  in  any 
example  in  division,  then 

(\)  a  =  Iq  -\-  r. 

From  this  we  see  that  a  common  factor  of  h  and  r  will 
be  a  distributive  factor  of  ^^  +  r  and  therefore  a  factor  of 
a ;  that  is,  A  common  factor  of  b  and  r  is  also  a  common 
factor  of  a  and  h. 

Again,  by  subtracting  bq  from  each  member  of  0, 

(f)  a  —  bq  =  r. 

Hence  a  common  factor  of  a  and  ^  is  a  distributive  factor 
oi  a  —  bq,  and  is  therefore  a  factor  of  r;  that  is,  A  common 
factor  of  a  and  b  is  also  a  common  factor  of  b  and  r. 

These  two  conclusions  establish  the  fact  that  b  and  r 
have  the  same  common  factors  as  a  and  b^  and  no  others; 
in  other  words,  instead  of  finding  the  H.  C.  F.  of  a  and  b, 
we  may  find  the  H.  C.  F.  of  b  and  r. 


210  ALGEBRA. 

In  Model  I,  therefore, 

the  H.  C.  F.  of  a  and  h  is  the  same  as  the  H.  C.  F.  of  r, 
and  h\ 

the  H.  C.  F.  of  r^  and  b  is  the  same  as  the  H.  C.  F.  of  r^ 
and  r^'y 

the  H.  C.  F.  of  r^  and  r^  is  the  same  as  the  H.  C.  F.  of  r^ 
and  r,; 

the  H.  C.  F.  of  r,  and  r,  is  the  same  as  the  H.  0.  F.  of  r^ 
and  Tg ; 

the  H.  C.  F.  of  r^  and  r,  is  the  same  as  the  H.  C.  F.  of  r^ 
and  r^. 

Since  r^  is  contained  in  r^  without  remainder,  the 
H.  C.  F.  of  r^  and  r^  is  rj  and  r^  is  therefore  the  H.  C.  F. 
of  a  and  l. 

229.  This  method,  if  applied  to  algebraic  expressions, 
very  often  requires  important  modifications. 

Model  J.— Find  the  H.  C.  F.  of  Ix^  -  Sa;^  -  Ho;  -  66 
and  3a;*  +  ^x^  -  llx  -  22 : 


q^=  2  a  =  7a!'^  -  3a;»  -  Wx  -  66 
6a;*  +  4.'c'-*  -  34a;  -  44 


ri=a;*  -  3a;»  -  4a;2+  28a;  -  22  9a;34-14a;«-86a;+  44  =  ra 


3a;H  2x^-  llx  -  22  =  5 
3a;4  -  9a;=*  -  12a;*-f  69a;  -  66 


3  =  g. 


Here  we  find  that  while  r^  and  r^  have  the  same  H.  C.  F. 
as  a  and  b,  their  quotient  will  contain  fractions  whichever 
is  used  as  divisor.  Going  back  to  first  principles,  we  may 
say  that  since  r^  does  not  contain  9  as  a  distributive  factor, 
we  may  introduce  it  as  a  distributive  factor  in  r^ ;  then  9^^ 
and  r,  will  have  the  same  H.  0.  F.  as  r^  and  r^.  Some 
such  device  as  this  may  have  to  be  used  several  times  in 
one  example,  or  in  rare  cases  may  not  be  needed  at  all. 


HIGHEST  COMMON  FACTOR. 


211 


gi=2 


^3=   X 


?4^41 


6ar*  +  4ic2  -  34aj  -  44 


7\  =  ^-  3aj3  -  Ax^  -f-  23aj  -  22 
9 


9aj4-27a53-36a;«+207aj-198 
Qaj-*  +  14a;3  -  86a;«  +  44a; 


r3=-41a;3+50x2+163aj  -  198 
-9 


369ic=^-450aj«-1467a;  +1782 
369a!3+574a;«-3526a;  +1804 


3ic44_2a?«-17a;-22  =  6 
3aj4_9a;3_i2a;5+69a;-66 


9a;«+14a;«-8Caj+44  =  r2 


3  =  ga 


r4=-1024a;2  +  2059a;  -  22 
1024a;5  -  2059a;  +  22 

Here  again  it  would  be  profitable  to  depart  from  routine : 
rather  than  multiply  r,  by  1024,  it  would  be  easier  to  factor 
1024^2  _  2059:c  +-  22.  We  find,  in  fact,  that  it  is  the 
product  of  a;  —  2  and  1024;:^  —  11;  and  only  the  first  of 
these  will  divide  r^\  hence  a;  —  2  is  the  H.  C.  F. 

230.  This  method  is  somewhat  more  perplexing  at 
times  than  the  method  of  elimination  of  highest  and  low- 
est terms;  and  the  following  cautions  must  be  carefully 
observed : 

I.  Remove  distributive  factors  before  beginning 
operations;  if  they  are  common  to  a  and  h,  restore 
them  at  the  end  of  your  work. 

II.  When  you  find  that  any  remainder  will  not 
serve  as  divisor,  carefully  look  for  distributive  factors, 
and  if  there  are  any,  remove  them;  THEif  introduce 
whatever  factors  are  necessary  into  the  last  divisor. 

III.  Continue  to  use  the  same  divisor  as  long  as 
possible. 

IV.  When  you  get  a  quadratic  remainder,  factor  it. 
All  the  examples  given  under  the   other  method  will 

serve  for  practice  in  this. 


CHAPTER  IX. 
LEAST  COMMON  MULTIPLE ;  SUMMATION  OF  FRACTIONS. 

231.  If  one  expression  is  a  factor  of  another,  the  second 
may  be  called  a  multiple  of  the  first.     Thus  x^  —  16  is  a 

multiple  of  x^  -{-  4:,  X  -}-  2,  X  —  2,  x^  —  4,  x^  —  2x^  -\-  4:X  —  S, 
and  x^  +  2x^  -\- 4.x -\- S, 

Again,  a;*  —  16  is  a  common  multiple  of  x^—  2x^  -{-  4:X  --  8 
and  x^  +  2x^  -}-  4:X  -\-  8,  just  as  192  is  a  common  multiple 
of  48  and  32. 

232.  The  lowest  common  multiple  of  two  or  more 
expressions  is  the  expression  of  lowest  degree  in  which 
those  expressions  may  be  found  as  factors. 

Thus  x^  —  4:  and  x^  -\-  4:  have  ^*  —  16  for  their  lowest 
common  multiple;  just  as  9  and  16  have  144  for  their 
least  common  multiple  in  arithmetic ;  x^  -\-  4:  and  x  -{- 2 
have  for  their  lowest  common  multiple  x^  -\-  2x'^  +  4:^  +  ^5 
although  x^  —  16  is  a  common  multiple;  and  x^  —  6x-\-  6 
and  x^  —  4:  have  x^  —  3x^  —  4:X  +  12  for  their  lowest 
common  multiple. 

EXERCISE  CVII. 

In  the  following  table  the  expression  in  the  third  column 
is  the  L.  C,  M.  of  the  corresponding  expressions  in  the  first 
two ;  find  out  hy  what  each  of  those  first  ttvo  expressions 
must  be  multiplied  to  give  the  L.  0,  M, 

212 


LOWEST  COMMON  MULTIPLE. 


213 


1. 

x^  —  dx  -^2 

x'^'-bX  +  4: 

2:^  -  72:2  +  14^  _  3 

2. 

x^-^x-lb 

2:2-25 

2:3  +  32:2  -  252:  -  75 

3. 

x^+x-^O 

x^  +  lOx  +  25 

^3  _!_  6^2  _  15^  __  iQo 

4. 

x^  -  27 

2;3+22;2+6a;-9 

2:*  -  2:3  -  27ic  +  27 

5. 

a;2  +  8a;+12 

a;2  +  52:  -  6 

2:3  ^  7^2  _|.  4^  _  12 

6. 

2a;2  +  cc  -  6 

22:2  _  5^  _^  3 

22:3  -x^-lx  +  Q 

7. 

x'^y  —  xy^ 

2:*^2  _  x^y^ 

x^y^  —  2:2y^ 

8. 

3a;2-a;-10 

22;2- 8 

62:3  ^  222:2  _  282:  -^  80 

9. 

^x^  -  9:^:  -  2 

252:2  _  1 

252:3  _  50^2  _  ^  _^  2 

10. 

lQx^-^x-% 

502:3  __  18^ 

1002:*-1202:3-362:2+542: 

Very  Simple  Denominators. 

233.  In  adding  fractions  it  is  most  convenient  to  use 
the  Lowest  Common  Multiple  of  the  denominators. 
Model  A. — Eeduce 

2:  —  5  _  22;-  13  ,  2^  _  5  +  22; 
"T~  6         ^  8  12 


The  product  of  all  the  denominators  would  be  a  common 
multiple,  but  it  would  be  a  very  large  number;  the  least 
common  multiple  is  24.  We  can  reduce  each  of  these 
fractions  then  to  24ths. 

The  first  fraction  has  to  have  its  denominator  multiplied 
by  8  to  give  24 ;  then  the  numerator  must  be  multiplied 
also  by  8,  in  order  not  to  change  the  value  of  the  fraction ; 
similarly  for  the  other  fractions ;  so  we  get 

82:  —  40      82:  —  52  ,   62;       10  +  42; 


24 


24 


24 


24 


which  reduces  to 


22:-  10 
24 


214  ALGEBRA. 

and  this  again  reduces  to 

V  X  —  b 

'•f  "^88— %.:;'■    ■  •  , 

EXERCISE    CVIII. 

hi  the  same  way  reduce : 

^'       3     "'"■"5  30~'    ^'       6  To      '       l5~' 

a?  -  ^  _  2:r^  -  a;  -  5  _  5  -  2a:  -  ^x^ 
^•4  6  12 

3a  -  ^  _  2^  -  7a  _  3a_ 
*•         8  ~~l0  20* 

x^  -  27       ^x^  -  5  +  ^'       -  x^  -  ^ 
^'5  3  "*"         21       • 

{x  -  2)(a;  -  3)       (2  -  a;)(3  -  x)  _  (:^;  -  2)(3  -  x) 
^'  7  "*"  21  "  14  • 

(3a;-5)(2a;+3)       (72;-l)(2:^;-l)  _  (a;- 3) (3 -a;) 
''•  2  "*"  6  18 

,(?£±^)(!£:^)_(.._3,)(?£^L^. 

^«-   I 18 )[-^)  +  l^  -  ^Jl 36 )•        • 

x-\-l      %x—l      a;^— 4  Sa:^        a;       1— a; 

^^-  "T"         6     "^  '^:^'         ^^-  ^^4  ^  i"  ~  "To"  • 

3a:^+7       2a;^-3  __  ^+1  x^-y'^  x^  y__ 

^^'     a;+5    "^  3a:+15  9    '  ^^'    x^y    +4a;+4^  +  *2' 

_  a;+l  _  5       2a;+3  a;^  +  7  x  -3a;-6 

^^'        a;-3      9       2a:-6'      ^^'  2a;-l  ""  4^^  "*"    6a:-3  * 


9. 


17. 


LOWEST  COMMON  MULTIPLE.  215 


3^:2     ,    a;2-3     ,    x-^  2  5.4 


20. 


x~-7   '  49-7a;   '     14  '  3:?;-2      32^-2  '   12-18^ 

5x  Qx 7x  dx^ 

X  —  y  ~^  Ix  —  7y      ^x  —  Sy      x^  —  y'^* 


Factoring  for  the  L.  C.  M. 

234.    The  L.C.M.  of  several  algebraic  expressions  may 
often  be  found  by  factoring. 
Model  B.— Find  the  L.  C.  M.  of 

,^2  _  9.    ^3  _  27;    ^2  _  4^  _|_  3.    ^4  _j_  9^2  _|_  81-    ^3  ^   27. 

Here  the  L.C.M.  must  contain  all  the  factors  of  any  one 
of  the  expressions;  we  find  the  factors  to  be: 

©2;2-  9  =  {x-{-Z){x-  3) 

(D  ^3  -  27  ^{x-  Z)(f  +  32;  +  9) 

(D  ^2  _  4^  _j_  3  ^  (^.  _  3)(^  __  X) 

0  ^^4  _|_  9.^2  _j_  81  =  (:^2  ^  3^  _|_  9)(^2  _  3^  _|.  9) 

®  ^^  +  27  =  (:?;  -f  i){x?  -  32;  +  9) 

The  L.C.M.  contains  the  factors  of  0;  then  to  make  it 
contain  the  factors  of  (2),  2:^  +  3^;  +  9  must  be  included, 
and  so  on;  including  in  the  L.C.M.  successively  the  factors 
of  each  number  that  are  not  already  in  it. 

Starting  with  the  factors  of  0,  the  L.C.M.  includes  as 
factors 

(x^Z)(x.^Z) 

Then  to  contain  ©  the  L.  CM.  must  also  include  (x?-{-Zx-^^) 
Then  to  contain  (3)  the  L.C.M.  must  also  include  (2;— 1) 
Then  to  contain  ®  the  L.C.M.  must  also  include  (x^—'6x-\-^^ 
Then  to  contain  (5)  the  L.C.M.  needs  no  more  factors. 


216  ALOEBBA, 

EXERCISE    CIX. 

Find  the  L.  CM,  of: 

1.  x^  -  121;  x'  -  2x  -  99;  6a^  -  55x\ 

2.  27x^-8;  36a;2-16;  6:^2  +  11:^-10;   9a;2  +  9a;  -  10. 

3.  x^  —  S;  x^  +  4:X^  +  16 ;  x'^  +  4.x^  -  Sx  -  8. 

4.  a^  +P;  a-  b;  a'  +  a^'  +  h\ 

5.  x^  —  x{a  -\-b)  -\-  ab]  x^  —  a^\  {x  +  a){x  —  by\ 

6.  x^  +  Sx^;  x^  -  9x;  x^  +  27 ;  {x  +  3)^  -  9x, 

7.  x^  —  y^]  x^  —  y^;  ^  +  ^• 

8.  x^^  —  y^;  x^  —  y^\  x^  —  y^]  ^  —  y\  x'^  -\-  xy  -\-  y^, 

9.  6x^  -  I82;  +  12;  4^3  _  16^  _|.  12 ;  2x^  -  10.^  +  12. 
10.  x^'-{y  +  zY;  y^-{x  +  zY;  z^-{x  +  yf. 

More  Complicated  Denominators- 
235.  Model  C. — Simplify  the  expression: 

3a;  -  1  3  ?>x-^l  3 


^3  _  27       x^-]-  9x'  +  81    '   x^+  27      x;^  -  9' 

The  L.  C.  M.  is 

{x  -  3)(cc2  +  3a;  +  9){x  +  3)(a;2  -  3a;  +  9)  =  a;«  -  729. 
The  first  denominator  must  be  multiplied  by 
(a;  +  3){a;2-  3a;  +  9) 
to  give  the  L.  CM.;  then  the  numerator  must  be  multi- 
plied by  the  same  expression : 

3a;  -  1  _  3a;^  -  x^  ^  8lx  +  27 

a;3  -  27  ""  x^  -  729  * 

Similarly  for  the  other  fractions.     The  entire  expression 
thus  becomes 

3a^-a;>-81a;  +  27  _  3a;^-27      Zx* -\- x^ -S\x  -  21  _  3a;<  -  27a;'^  -  243 
a;«-729  a;-^-729"^  a:«  -  729  a;«  -  729 

_  3aj*  +  24a;2  -  162a;  +  270 
x""  —  729 


LOWEST  COMMON  MULTIPLE.  217 

EXERCISE    ex. 

Simplify  the  expressions : 

3a;  7 2_ 

^-  4ic2-9  +  2a;  +  3      2x-d' 

2_+_^      ^_-3  1  -a; 

2-  ^'3  _  4  +  3^"^r6       8:c  +  16* 

X  3.4 


^3  _  27       a;2  _  5^  _|_  6   '    3^.2  _^  g^  _^.  27' 

a;  +  1  1  —  x  a:  —  2  2  —  a: 

2a;2  -  18  "  x^-4:X^Z  "^  a;^  _  1  ~  x^j^'ix-  3* 
2  a;2-l        _a;2  +  a;  +  l  2a; -3 


3   '   :?;'^  _  7a;  +  10      x^  —  3a;  -  10       3a;  -  15' 

d^  +  ^^   ,  ci h__        a  +  h 

^'  a^-h^'^  d'  +  ah  +  l?'      a'-ai'^  ah-V' 

{x^  -\-  y^Y  x^  +  y^  x^ 


x^  +  x^y^  -\-  y^      x^  —  xy  -\-  y^      x^  ~\-  xy  -\-  y^' 

3a;  +  2       __  2a;  —  3  a;  +  2 

^^^"3a;  -  28       a;^  -  16  "^  x^  -  11a;  +  28' 

2a;2  +  3    ,        7a;  .      2a;       _ 

~5a;  +  15  ~     ' 

a; +  3'       . 

Fractional  Equations. 

236.  An  expression  which  forms  one  entire  member  of 
an  equation  may  have  its  value  changed,  without  destroy- 
ing the  equation,  provided  the  other  member  of  the  equa- 
tion also  has  its  value  changed  in  the  same  way  and  to  the 
same  extent;  but  we  must  be  careful  not  to  change  the 
value  of  an  expression  which  is  not  a  member  of  an  equa- 
tion. 


218  ALGEBBA, 

Thus  in  the  equation 

"^        '        7       ~     14  ~4         ^28 

we  may  ^^ clear  of  fractions^'  by  multiplying  by  28;  but  in 
the  expression 

x  —  b       3a;  —  2 
~ir  ^        7 

we  can  not  clear  of  fractions ;  we  have  no  right  to  multiply 
by  14  or  by  anything  else;  we  may  reduce  to  one  fraction 

7a;-  5       6a;  -  4  ^  13a;-  9 
14      "*        l4~  ""       14 

but  we  cannot  lose  sight  of  the  denominator. 


EXERCISE    CXI. 

Simplify  the  following  equations : 

X  —  1       3a;  +  2  6      _  x'  —  x 

^'   "~2        ^         6         '~  a;  -  3  ~  x^^' 
4a;  +  7    ,    3a;  -  4       20a;  -  3 

5       ^      15  15 

4a; -3^     3x+7y     6x-2y     9a;  +  2i/_3?/     2(a;+l) 


3. 


i 


14  21       '       42  14 


1,1  8  -  a; 

4.    Z a  + 


6. 


x  —  6       a:  +  5        (a;  —  6)  (a;  —  5) ' 

1 2       _        1 

1  —  a;       1  —  a;^  ~  2a;  —  5' 
16a;  -  x^  _3  +  2a;_2-3a;_l 
a;2-4  a;-2    ~  Y+x'  ~  8  ' 


1  1 


a;2+9a;  +  20   '    a;2+12a;+35       a;2+lla;  +  28 


LOWEST  COMMON  MULTIPLE.  219 


8. 

1 

1 

x' 

— 

13a; 

+  42    ' 

9. 

1 

1 

x^ 

+ 

Ix 

-44   ' 

10. 

~^ 

__ 

1 

-T-1T  + 

x^  —  Ibx  +  54       x^  —  l^x  +  63* 

1 _  1_ 

x^  —  '^x  —  143  ~  x^  —  llx  +  68* 
2^  1  a;-l 


:?;2  +  4rr  +  3  ^  iz;2  _j_  5^  _|_  g  -  ^_|.  3' 

The  Three  Principal  Signs. 

237.  The  entire  numerator  of  an  algebraic  fraction  rep- 
resents some  number  which  may  be  +  or  —  according  as 
one  set  of  values  or  some  other  set  is  assigned  for  the  letters 
appearing  in  it;  so  also  the  denominator  of  an  algebraic 
fraction  may  be  +  or  —  ;  and  the  quotient  represented  by 
the  fraction  would  have  its  sign  +  or  —  according  as  the 
entire  numerator  and  the  entire  denominator  had  signs 
alike  or  unlike. 

In  the  expression  x  — the  values  a=  2by  b  =  8, 

q  ~  5  give  to  the  fraction  the  value  -\<^-  =:  10,  so  that  the 
value  of  the  expression  is  a;  —  10. 

The   fraction  is  unchanged  if  it  be  written -,  be- 

q  —  b 

cause  in  this  case  we  have  changed  the  signs  of  numerator 

and  denominator  both. 

If,  however,  we  write  the  fraction  ~ ,  its  value  becomes 

b-q 

3Q 

— ^ —  =  —  10;  and  if  we  then  wish  the  value  of  the  ex- 
pression to  be  unaltered,  we  must  change  the  sign  of  the 

term  in  which   the   fraction  appears.     Thus   x  —  % 

b  —  q 

would  be  ic  —  (—  10)  =  a;  -f  10;   to   keep  the  entire  ex- 
pression unchanged  in  value  we  must  write  \i  x -\- 1 . 


220  ALGEBRA, 


So  also  if  we  write  the  fraction  ■ ^,  then  we  must 

q  -I 

write  the  expression  x  + 


q-V 

238.  Whenever  an  algebraic  fraction  appears  as  a  term 
in  an  expression,  it  is  convenient  to  recognize  as  the  THREE 
PRINCIPAL  SIGNS  of  the  fraction : 

I.  The  Sign  of  the  Whole  Numerator ; 

II.  The  Sign  of  the  Whole  Denominator ; 

III.  The  Sign  hefore  the  Fraction  ; 

and  any  two  of  the  three  principal  signs  of  a  fraction  may 
be  reversed  without  altering  the  value  of  the  expression  in 
which  the  fraction  appears. 
Model  D. — In  the  expression 


X  —  \      \  —  x^   ^   \  -\-  X 

it  is  desirable  that  the  second  denominator  should  be 
written  o;^  —  1  so  as  to  bring  out  more  clearly  the  fact  that 
the  first  and  third  denominators  are  factors  of  it.  The 
expression  may  be  written 

x^\^  x^  —  \^  x-\-\ 

without  altering  its  value. 
Model  E 

x-\-l  "Ix-l  ,  2  -  3:z: 


(x  -  3)  (:^;  -  5)       (x  -  2)(3  -  2:)  ^  (2  ^  x) (5  -  x) 

may  be  written 

xJ^"^  2a; -1  32^-2 


(x  -  Vi(x  -  5)  ^  (ir  -  %){x  -  3)       (2:  -  2)(a;  -  5) 
without  changing  its  value;  as  follows: 


LOWEST  COMMON  MULTIPLE,  221 

in  the  first  fraction  there  is  no  change; 

in  the  second  fraction  the  sign  of  one  factor  of  the 

denominator  is  changed;  this  changes  the  sign  of  the 
whole  denominator,  which  is  one  of  the  three  principal 
signs;  so  the  sign  of  the  whole  fraction  is  also  changed; 

in   the  third   fraction  we  change  the  sign  of  two 

factors,  which  leaves  the  sign  of  the  whole  denominator  un- 
changed;  the  sign  of  the  numerator  is  changed,  so  the 
sign  of  tlie  whole  fraction  must  also  be  changed. 


Rearrangements. 

239.  The  altered  form  of  the  above  expressions  will  be 
found  much  more  convenient  in  the  process  of  uniting  the 
three  fractions  into  one.  The  principle  of  arrangement  in 
this  case  is  to  have  the  ic-term  in  each  binomial  come  first, 
so  that  one  may  apply  mentally  the  process  of  cross- 
multiplication. 

240.  There  is  another  principle  of  arrangement  which 
it  is  sometimes  expedient  to  follow.  It  is  known  as 
cyclic  order,  and  is  of  importance  to  all 

students  of  Algebra. 

According  to  this  principle,  if  we  had 
four  letters  a,  h,  c,  d,  we  should  make  h 
follow  a,  c  follow  h,  d  follow  c,  a  follow     ^  ^ 

dy  h  follow  a,   and   so  on;    the  several       ^  v^^^^^ 
letters  taking  turns  and  beginning  again 
at  the  first,  as  if  they  were  arranged  in  a  circle.     Thus  the 
expression 

(a  -  1){1)  -  c)  -  (Z>  ~  c){a  -c)-\-(h-a){c-  l) 
would  be  written,  according  to  this  principle,  thus : 

{a  -  l)){h  -c)  +  {'b-  c){c  --a)  +  {a-  h){b  -  c) 
without  change  of  value. 


222  ALGEBRA, 


EXERCISE   CXII. 


Simplify  the  following  expressions  : 

1,2,3 

•  X  -  2^  X  +  2^2  -  X 

^*  dx+l'^  9x^-  l~^l-  dx 

3  iz:        .    1  —  a; 

»  I   i 


1  -  4:?;2  '   2:^-1   '^  22;  +  1* 

1    +,^,^^_^+ 


(ii;  -  2){x  -  3)   ^    (:^;  +  2)(x  -  3)  ^  4  -  :z;2- 

2 3  5 

i^;2  _  7^  +  12       12  -  a;  -  a;^  +  48  -  32;2' 

c  ,  a  c 


"•  («  -  ^)(«^  -  c)^  {c  -  h){c  -  a)^  {h  -  a){h  -  c)' 

7.  7:7— w^_-^  -  /TT—Twr— :;^  + 


10. 


11. 


{a  -  h){h  -  c)       {h  -  c){a  -  6^)  ^  («  ~  c){b  -  a)' 

a  -^  h  b  +  c 

{b  -  c){a  -  c)  +  {b  -  aj{c  -  ay 

{a  -  b){c  -a)^  {c  -  b){c  -  a)^  {b  -  a){b  -  c)' 

c? b^ 

{b  -  a){b  -  c)  +  (a  -  b){a  -  c)' 

a  2b  2b 


{b  —  c){c  —  a)       {a  —  b){a  —  c)       {b  —  c){b  —  a)' 


a;                ,               5  ,  5 

12.   JZ K^l7. ^  +  77. ^u7. :7^  + 


13. 


(^  -  b){y  -x)^{y-  b){y  ~  x)^  {b  -  x){b  -  y)' 

X 10 10 

(5  -  y){y  -x)"  (x-  b){x  -  y).  ~  (5  -  y){b  -  x)' 

*  Find  the  shortest  way  of  doing  this  example. 


14. 


LOWEST  COMMON  MULTIPLE.  223 


(5  -  y){x  -  y)       (5  -  x){y  -  x)' 


x^  y^ 

^^'  {y  -  ^){y  ~  7)  "^  (^  -y){x-  ly 

If  y  ^'i  in  the  following  expressions,  what  value  must 
X  have  m  each  case  to  make  the  whole  expression  equal 
to  f  f 

^1  1        ,       2x 


X  -\-  y  X  —  y      x^  —  y^' 

*i7  ^  ~-  y  I  y"^  +  ^^y  .  ^  +  y^ 

'  X  -\-  y        y'^  —  x^        X  —  y 

*    a;  +  ^  y^  —  x^^  X  —  \ 

x^  —  y'^      xy  —  y'^ 


19. 


*on  x^  +  y'     _j^  +-^_. 


Solve  the  following  equations  : 

x^  —  bx  2 

21. 


22. 


n;^  —  4rc  —  5       3 
3a;^  +  6a;      _  7 

ii;2  +  4:^;  +  4  "■  3  * 

2;2  -f  5:^;  _|.  6    i?;2  _  2a;  —  3 


=  5. 


2a;2  +  13^  +  15       2x^  +  11a;  +  5       ^ 

04.      1 ! ,  -i-  ■ — '         ! z=.  5 

2a;2  -  2;  -  1        4a;2  +  a;  -  14  _  1^ 
^^'  2x^  +  bx  +  2  ^      l(Jx^  -  49     ""5"' 

*  First  combine  tLe  fractions  whose  denominators  are  of  the  first 
degree. 


224  ALGEBRA, 

1      .  1  1 


26. 


27. 


cc+l  '  {x-\-l){x-^2){x-\-^)~ {x+l){x-\-'^)~ ^-x 
3  + 2a;      2  — 3a;      16a;  -a;^       1 
"2  -a;  ""  2  +  a;  +    a;^  -  4   ~  II' 

a; 3  1  _2^ 

^®*   (a;-3)(a;+l)       (:i;-|-2)(a;-3)  +  (a;+2)(a;+l)~"3* 

^^-  (3-a;)3(a;-2)  +  (2-a;)3(3-a;)2+(5a;-6-a;2)(a;-2)'^^' 
*  ^  '^  4-20a;_a:-2      a;+2         2 


31. 


l-2x      l+2a;       4a;2-l  '~a;+2'^2-a;      4-a;2* 

1 + L__- ^ 

a;2  -  9a;  +  20  ^  a;2  -  11a;  +  30  ~  ll(a;  -  4)' 
1.1  4 


:  + 


•  a;2  -  7a;  +  12  ^  5a;  -  a;^  -  6  ~  5(3  -  a:)(4  -  x)' 

33    ^  ,  1  .  ^  -0 

•  2a;2  -  a;  -  1  "^  3  -  a;  -  2a;2  "^  2(2a;  +  1)(1  -  x) 

1  3  1 

35 


1  +  a;  -  2a;2  "T~  6a;2  -  a;  -  2       (2a;  +  1)(1  -  a;)* 

2  .        3  3  -  2a;  1 


36. 


37. 


X    '   1  -  2a;  ^  4a;2  -  1       a;  —  2a;2' 

113  a;-l 


a;  -  1      a;  +  2       (x-\-2f~{x-  x^){x'^  +  ^  -  2) 

5        _         1 24  5 

2(a;+l)       10(a;-l)       5(2a;  +  3)  ~  2a;2  + 5a;+ 3* 


5  2  3a; 


39. 


5  +  a;  -  18a;''^      2a;2  +  5a;  +  2  ~  (2  +  a;)(5  -  9a;)* 
11,1  12 


x+l    (a;  +  l)(:r+2)^(a;+l)(a;+2)(a;+3)    (a;+l)(a;+3)- 
*  Simplify  eacli  member  separately  before  clearing  of  fractions. 


LOWEST  COMMON  MULTIPLE,  ^25 


Modified  Methods  of  Reduction. 

241.  There  are  certain  types  of  fractional  expressions 
and  equations,  of  frequent  occurrence,  which  can  be  much 
more  easily  reduced  by  modifying  somewhat  the  ordinary 
straightforward  method.  For  instance,  examples  12,  13, 
14,  16,  and  26  in  the  preceding  exercise. 

Model  F. — Reduce 

1.1.         6  54 


3-^'3  +  a;'    9  +  ir2    '81  +  a;^* 

In  this  example,  if  we  combine  the  first  two  fractions, 
then  that  result  with  the  third,  and  finally  that  result  with 
the  fourth,  the  work  can  all  be  done  mentally. 

Sometimes  the  denominators  are  seen  to  belong  together 
naturally  in  pairs: 

Model  a. 

3  2  3  2 


2cc—  3      x  +  l      "^x  +  ^   ^  X-  1 
18  4 


■  {2x  -  3)  {^x  +  3)  ^  (x+l){x-  1) 
18rr2  -  18  +  16a;2  -  36  Zix^  -  54 


-  .   {4:3?  -  •d){x'  -  1)      -  4a;*  -  13a;^  +  9" 

Model  H. — Reduce 

x-%^  X- 3      x-&^  x-1 

1            1            1            1                        1 

^  x-2      x-l~'x-&      x-Z         ^      x-3 

1 

x-7 

r                                                        q 

®  ^  _  9^  +  14   -  ^2  _  93;  ^  18      °^°'°  ^"  ® 

226  ALGEBRA. 

®  -  5{x^  -  9x+18)  =  d{x^  -9x  +  14) 

®X{x^-9x  +  U){x^  -9x  +  18) 

(§)  -  5x^  +  4:5x  -  90  =  3x^  -  27^;  +  42      same  as  ® 

®  8a;2  -  72:?;  +  132  =  0  (5)  +  5x^  -  45^;  +  90 

®  2:?;2  _  x3^  ^    33  ^  0  ®  -^  4 

Whence  by  the  Quadratic  Formula  x=  3.218;  x  =  1.718. 
The  advantage  of  rearranging  the  equation  as  in  @  is 
that  the  a;-terms  destroy  each  other,  and  we  save  such 
multiplications  as 

{2x  -  6){x^  -  13:?;  +  42)   and   {2x  -  ld){x^  -  6x  +  6). 

Model  I. — Keduce 


X  —2      X  —  1  __x  —  6      X  —  5 
x-3  "^  x-2  ~  a;—  7  "^  x~^^' 


K  we  divide  the  numerator  of  the  first  fraction  by  its 
denominator,  the  quotient  is  1  and  the  remainder  1;  so 

that  the  fraction  reduces  to  the  mixed  number  1  -\ -. 

X  —  3 

The  rest  of  the  reduction  proceeds  as  in  the  preceding 
example. 

Model  J. — Keduce 

x^  +  7x  +  15  __        2x^  +  3a;  +  3 


2x^  +  13x^  +  24tx  -  10      4a;3  +  42;^  +  5a;  -  4 


LOWEST  COMMON  MXILTIPLB,  227 

The  reciprocals  of  these  two  equal  fractions  are  equal, — 

^         x^+lx+lb         ~      2a;2+32;+3  '  ^ 

Eeducing  to  mixed  numbers,  by  carrying  out  the  division 
indicated  by  each  fraction, — 

(3)2^-1+    .^j;   ,^^=2^-1  + 


x^  +  lx  +  lb  ^  2ii;2  +  3a;  +  3 

same  as  © 

^x^-^lx  +  lb_^x^  +  Zx  +  ^ 

^         ^^75         -       2^-1  ^-^ 

5  5 

(D  a;  +  2  -J -—  =  a;  +  2  +  r same  as  (5) 

a;  +  5  2a;  —  1 

Whence  a;  +  5  =  2a;  —  1  and  a;  =  6, 

Care  must  be  taken  not  to  apply  the  method  of  the 
example  just  preceding  to  equations  where  there  are  more 
than  one  term  in  either  member;  that  is,  if 

3        ,         a;-3        _       a;3  +  3 


a;  —  2      a;^  —  5a?  +  1      a;^  +  5a;  +  2 

IT   IS   NOT   TRUE   THAT 

a;  -  2      a;^  -  5a;  +  1  _  a;^  +  5a;  +  2 
3"^        a;-3        ~       a;2  +  3' 

In  examples  like  number  1  of  the  next  exercise  clear  of 
the  numerical  denominators  first. 


22S  ALGMBMA, 


EXERCISE    CXIII. 


Reduce  to  the  simplest  form  : 

11a;  -  7       5x  -  7  _  7a;  -  3       dx  -  5 
^'        12        +       3       '-  X  -  1  ^       5 
1,1  4a;        .  17 


2a;  -  3   '   2a;  +  3      4a;2  +  9  ^  16a;*  -f  81' 


3a; 


_  3a;^  +  2         4(3a;  +  2) 

1  Q^2      I       Q      I        oi  ^4     I      1  /?   ' 


^-  6a;  -  4      3a;  +  2       ISa;^  +  8  ^  81a;*  +  16 
3  7  2  4 


6. 


2a;  —  3'3  +  a;'2a;4-3       3  —  a;* 

3  1  x  +  4:  x  +  1 


x  +  1       2(a;  +  2)       (^  + l)(a;+ 2)  "^  (a;  +  2)(a;  + 3)' 


1  4,5  4,1 


X  —  3      X  —  1      X       X  +  1      X  +  d' 
2a;  +  1       4a;  —  7       x  —  6  _:?:-f-53a;  —  2 
''*         5         ^  a;  -  4  +       3      ~  2a;  -  8  "^         3      * 
a;  -  1   ,   20a;  +  11  _  a:  -  11       41a;  ^  4 


9. 


a;  —  3   '  3  3a;  —  9   '  6 

1.2  2.1 


4  —  a;'   2a;  —  3       7  —  2a;'a;  —  1* 


1,1  1  1 

10.  -  + 


a;5— -a;      x  +  1      4—  a;' 

2  2  2  2 


11. 


a;— 2      a;  —  7       x  —  1       6  —  x 


1,1  1,1 


13. 


14. 


a;  +  9'a;  +  5      a;  +  8a;  +  6* 

3a;2  —  a;  —  25  _  x^  +  7a;  —  1 


9a;3  _  6^2  _  >^i^  j^  18       3^3  _^  20a;2  -  10a;  +  6' 

4a;^  -  4a;  -  3  _  2x^  -f  3a;  +  4 

4a;3  +  8a;2  -  10a;  -  3  ""  2^  +  9a;2  +  13a;  +  15* 


15. 
16. 


LOWEST  COMMON  MULTIPLE.  S2& 

Ix^  +  10:^-1  _  ^x^  +  ^^^  +  32 

llx^  +  103^2  -Ux-S  ~  33x^+  239x^+  410:?:  -  23* 

Qx^  -  X  +  13  _        4:x^  +  12a;  +  20 

6^3  +  lla;2  +  14:?;  +  24  "  ^x^  +  20^^  _|.  45^  _|_  45- 


3a;  +  13       x  -1  _4:X  +  17      Sx  +  d 

^'^'     x  +  3^x  +  l~~2x+Q'^4:X  +  4t' 

3a;  -  8   ,   4a;  -  17       2a;  -  13    ,    5a;  ~  41 

18.     TT  H 7-  = ^  + 


19. 


21. 


a;  —  3         a;  —  4  x  —  7  x  —  8 

8x  -  27       2a;  -  9  _  6a;  -  8  _  3a;  -  17 
2a;  -  7  "^  a;  -  5        2a;  -  3  " 'a;  -  6  * 
15a;  -  2       12a;  --  41  ___  14a;  -  19       4a;  —  8 
3a;  —  1   "^  3a;  —  11  ""    2a;  -  3    "^  2a;  -  5* 

1 1_  _       2a ^__ 

X  —  a      X  -\-  a      x^  -\-  a^      x^  -\-  cC 


23. 


27. 


,4' 


a;+4  "^  ^-^[6  ~  a;  —  5      a;  —  7' 

4a;  — 3  _  3a;  +  4      a;  —  5  __  3a; +  4  __  10a;  —  4         ^ 

66a;  -  49      4a;  -  5  _  14a;  -  19      24a;  -  43 
6a;  -  5    +  a;  -  1  ~    2a;  -  3    "^    3a;  -  5  ' 
x^  -f  28a;  -  55  _  3a;^  +  20a;  -  47 

7a;3+  201a;2-  244a;—  245  ~  21a;3+155a;2- 226a; -209* 

^■2  -t-  2a;  -  3        _        67a;  +  133 
^3  _[_  3^2  _  3^  _  3  -  67^2  ^  200a;  -  1' 

a;^  +  a;  -  6  +  1       8a;^  -  18a;  -  9 
x-2  +         4a;  +  1 

_  24a;^  -  55a;  +  26  1 

~  8a;  -  13  *    2a;  -  2* 

2a;^  +  21a;^  -  2a;  +  120  __  4a;^  +  6a;^  -  2a;+  105 
x^  +  lla;2  +  64        ""        2x^  +  3x^+56      ' 


230  ALOEBRA, 


The  Principal  Dividing  Line. 

242.  A  fraction  in  which  the  numerator  or  the  denomi- 
nator or  both  contain  a  fractional  term  is  called  a 
complex  fraction. 

For  example,  the  ratio  f  :  5  may  be  written  as  a  com- 
plex fraction;  so  also  2  :  |.  When  so  written  these  ex- 
pressions cannot  be  distinguished  except  by  the  Principal 
Dividing  Line  of  the  fraction. 


^,3        2        ,  .,    2       10 
Thus  r-  =  — ;  while  ^  =  — . 
5        15  d         3 

5 

243.  In  a  complex  fraction  everything  above  the  princi- 
pal dividing  line  constitutes  the  dividend,  everything  below 
constitutes  the  divisor.  Plus  and  minus  and  equal  signs 
which  precede  or  follow  a  complex  fraction  must  be  on  the 
same  level  as  the  principal  dividing  line  of  the  fraction. 

244.  In  most  cases  the  readiest  way  to  simplify  a  com- 
plex fraction  is  to  take  advantage  of  the  principle  that 
multiplying  the  numerator  and  the  denominator  by  the 
same  number  does  not  change  the  value  of  a  fraction ;  and 
to  choose  such  a  multiplier  as  will  render  both  num'erator 
and  denominator  integral. 

Model  K. 

.^•^  _     x^  —  if'    _x^  -\-  xy  -\-  y^^ 


x^  —  y^     x^{x^  —  y^)         x^{x  -f-  y) 

245.  Sometimes  one  prefers  to  simplify  the  numerator 
and  the  denominator  separately,  reducing  them  each  to  a 
single  fractional  term,  and  then  dividing  the  numerator  by 
the  denominator. 


LOWEST  COMMON  MULTIPLE,  231 

Model  L. 

a       l      aq  —  Ip 

V  ^  —  M  _aq  —  Ip  Ip  —  aq  _  ah 
P  ^  ~ pLzL^  ~  P^  '  ah  ~~  pq 
ah  dh 


EXERCISE  CXIV. 


Simplify : 


HA  5.      ''         " 


3 

27  "^    8  9. 


x^  —  y^'  '  9a^  —  46'^' 

.  b.    •  10. 

x___p_  \_       1        13 
a       h 


X 


y 


X       a 
y  ~~  h 


x  —  2 


x^-9 


11. 


M 

X  - 

-  5 

1  4- 

20  - 

-  9a;' 

^  n^ 

a;^ 

ax  - 

■0? 

^« 

a"  - 

■x' 

x^- 

8 
4 

x'- 

X  — 

8 
4 

1 


X  ■ 
X 


3       5  8-  ^2       5^  _,_  g.  ^2    ^ 


L_^  a;2-9  ^3__1 

ic     6  y 

13.  ^  ~  ^.  *  ,  2z 


a 


1  + 


111 

- — ^—-—^ —  a       0       c 

'  b    ^  c    ^  a 


232  ALOEBBA. 


17. 


^  +  ^  +  3- 
1         5^       3^ 

"T"  Q^  +  /^ 


19. 


18. 


a;  +  3 

2x+l 

x-d 

'   2x-l 

x  +  3 
2a; +1 

x  +  y^ 

X  —  3 
2a;-  1 

a  ~  h 

x^y 

a  +  h 

a  ^  J) 

5    '   3a;' 

a—  b  X  --  y 
a+  h  a  —  h 
x+y      x—y  X  —  y~x  +  y 

Continued  Fractions. 

246.   Complex  fractions  of  the  following  peculiar  type 
are  called  continued  fractions  : 

Model  M. 


3  + 


2  + 


1 
4  - 


4-i 


To  simplify  this  expression,  rewrite  it  as  far  as  the  last 
dividing  line  which  indicates  a  complex  fraction ;  substitute 
for  that  its  simplest  form,  and  repeat  the  same  process  as 
often  as  necessary. 


3  H 3  H ^ — 


4-i 


398      199 


LOWEST  COMMON  MULTIPLE,  233 

EXERCISE  CXV. 

Simplify  : 

1  '1  .  1 


1. 


X 
4.    ? 5..  1  +  ^ 


1+  T-  ^  + 


'^-— 1  1  +  2-3-^ 


a 


1 


2a      25  j^  _      1 


25      2a  ,1 

*■                      5  '-^ 

^^-,^2a-3  '•  T- 

+          2a  ,    ,        1 


1-^  1  + 


1  +  -^ 

X 

8.  In  the  expression  substitute  for  x  the  expres- 

2  4-- 

sion — ,  and  reduce  to  simplest  form. 

'  +  4  +  1 

X      X 

9.  Find  the  value  of  the  expression  —  :  —  when 

ic  =  1  H -—  and «/  =  1 — ; 

'  +  3+1  ^  +  3^i 

a;'  is  obtained  from  x  by  omitting  the  fraction  indicated  by 


234  ALGEBRA. 

the  lowest  dividing  line,  and  y'  is  obtained  from  y  in  the 
same  way. 

10.  Find  the  value  of  the  ratio  —  :  — -,  when 

y   % 

,        1 


y  is  the  result  obtained  by  substituting  c  -\-  —  ior  c  in  the 

expression  for  a;;  ^'  is  the  result  obtained  by  interchanging 
h  and  c  in  the  expression  for  x,  and  ^'  is  the  result  of  the 
same  change  in  y. 


Another  Application  of  L.  C.  M. 

247.  If  we  have  an  equation  in  the  form  of  an  integral 
expression  equal  to  zero,  the  answers  can  be  found  if  the 
equation  can  be  factored.  The  answers  of  two  or  more 
such  equations  may  all  be  found  in  the  equation  formed  by 
setting  the  L.  C.  M.  of  the  expressions  equal  to  zero. 

Model  N. — The  equations  following  have  the  answers  set 
down  beside  them : 


x^—^x  =  0  X  =  0 

x^  —  bx  -\-  ^  =  0  X  =  2 

a;^  —  9  =  0  X  =  3 


X  =  3 
X  =  d 
X  =  —  3 


The  L.  0.  M.  of  the  expressions  x^  —  3x,  x^  —  5x  -{-  6, 
and  x^  —  9,  is  x{x  —  S){x  +  3){x  —  2) ;  and  the  equation 

x{x  -  3){x  +  3){x  -  2)  =  0 

has  answers  0,  2,  3,  —  3. 


LOWEST  COMMON  MULTIPLE,  235 


EXERCISE    CXVI, 

For  each  of  the  following  sets  of  equations  construct  a 
neio  equation,  among  whose  answers  can  le  found  all  the 
answers  of  each  given  equation;  and  let  the  resulting  equa- 
tion he  the  one  of  lowest  degree  which  can  satisfy  this 
condition : 


1. 

2a;2  =  3:?;  +  2;     4^^2  =  1;     x^ -\- 2  =  ^x. 

2. 

8:?;3_27  =  0;     4cX  =  - 

X 

3. 

4:x^  =  ±  I;     2a;  =  1. 

4. 

x^  =z  X  +  20;     x^  =  12  —  x;     x^  =  Sx  —  15. 

5. 

a;2  =  l(x  +  1);     x'^  =  Z{x  +  1)  +  J;     x^  =  ^x  - 

-  12, 

6. 

x^  =  8^3;     x  +  2a  +  ^^^  =  0;     x  =  ^^\ 

7.  (3a:  -  2)  :  (:r  -  1)  =  5  :  2x;    Qx^  +  5  =  Ux;    ^x^  =  1. 

8.  6x^=x+2;    lSx{l-x)  =  ^x^+x+2;     5a;(3^+l)=:a;+l. 

9.  x^  =  27;     x^  =  15a;  —  36;     x^  -  ^x^  -  2x  +  Q  =  0. 
10.  5a;2  +  19a;  =  4;  lOa;^  +  19a;  =  3  +  6a;; 

15a;2  +  19a;  =  13a;2  +  4(2a;  -  3). 

Tests  for  Simple  Factors. 

248.  Sometimes  it  is  not  possible  to  factor  all  the  ex- 
pressions by  inspection;  and  in  such  cases  it  often  serves 
to  get  the  factors  of  one  expression  and  to  try  them  as 
divisors  of  the  other  expressions. 

Model  0. — In  the  expressions 

2a;2  +  7a;  +  5;     2a;3  +  a;  +  3;     a;^  -  6a;  -  5 
the  factors  of  the  first  expression  are  (2a;  +  5)(a;  -f-  1);  of 


236  ALGEBRA, 

these  2iz;  +  5  is  evidently  not  contained  in  either  of  the 
other  expressions;  by  substituting  2;  =  —  1  in  the  other 
expressions,  we  reduce  each  to  zero,  and  hence  conclude 
that  :?;  +  1  is  a  factor  of  each.  Dividing  we  find  the 
factors  as  follows : 

2^3  _|.  ^  ^  3  =  (^  _|_  i)(22;2  -  2a;  +  3); 

x^  -  Qx  -  h  =  \x  -{-  l)\x^  -  X  -  b). 

Hence  the  L.  C.  M.  would  be 

{x  +  l){2x  +  5)(2^2  _  2a;  +  3){2;2  _  a;  -  5). 

EXERCISE    CXVII. 

Rearrange  the  equations  in  each  example  so  that  each 
shall  le  an  integral  algebraic  expression  equated  to  zero. 
Then  find  the  L,  G,  M.  of  their  first  members  : 

1.  2;2  -  4  =  0;     x^  +  X  =  10. 

2.  x^  —  7x  -  30  =  0;     2x^  +  15  =  13a;;     3x^  =  29a;  +  5. 

3.  2a;2=9a;+5;  a;(a;"2-9)=:20(a:-l);  a;2(2a;-l)  =  5(2a;2-5). 

4.  a;2  =  ^(a;  +  10);    a;(a:2  -  2)  =  |(a;  -  4) ;    Sa;^  +  4  =  7a:'^. 

5.  a;-2  =  fa;2;   a;^  -  12  =  15a;2 +  0;;    a;^  =  16  +  2a;*'^  +  So^^^ 

6.  a;2=|(9a:+4);   o;^- 6a;2+ 32  =  0;    ISa;^  -  19a;2  +  4  =  0. 

7.  a;3  =  8;     x^  +  4a;2  +  16  =  0;     x^  =  x^  +  4. 

8.  lOOOy^  =  1;     lOy^  =  99 f  -  1. 

9.  2a;3  -  6a;2  +  3a;  =  9;     20:^  z=  19a;  -  3. 

10.  2a;3-5a;2=16a;-40;  a;4-6a;2=:a;3-8a;+ 16;  2a;3=:a;2-j-25. 

Relation  between   L.  C.  M.  and  H,  C.  F. 

249.  The  L.  C.  M.  of  any  two  expressions  may  be 
obtained  by  dividing  one  of  them  by  their  H.  C.  F.  and 
multiplying  the  quotient  by  the  other. 

For  since  the  factors  of  the  H.  C.  F.  are  contained  in 
each  expression,  the  first  expression  contains  all  the  factors 


LOWEST  COMMON  MULTIPLE.  237 

of  the  second  except  those  that  are  not  included  in  the 
H.  C.  F. 

Model  P. — Of  the  two  expressions  2x^  —  ll:?;^  —  9  and 
4^5  _|_  11^4  _[_  gx  the  H.  C.  F.  is  iz:2  _^  2^  +  3;  the  quotient 
obtained  by  dividing  the  first  expression,  ^x^  —  llcc^  —  9, 
by  the  H.  C.  F.,  ^r^  +  2;r  +  3,  is  2^^  _  4^'^  _|_  2a;  -  3;  so  the 
L.  C.  M.  is  (22;3  -  4:x'^  -f  2a;  -  3)  {^x^  +  ll:r*  +  81) ;  if  we 
divided  the  second  expression  by  the  H.  C.  F.  we  should 
obtain  for  the  L.  C.  M.  {4tX^-^dx^-l%x-\-'il){2x^-llx^-^)] 
in  either  case  the  product  would  be 

8a;8+6:z;^-36x6+10^5-33:r4+162a;3-324:c2+162:c-243. 

EXERCISE    CXVIII, 

Fiyid  the  L.  C,  M.  of  the  following  expressions : 

1.  x^^  +  x^y^  +  y^^]  x^^  4-  x^y^  +  x^y^  +  x^y^  -\-  y^^. 

2.  x^+x  +  Q;  x^  + 2x^  +  9. 

3.  2x^  +  66xy^  —  20y^;  9x^  +  SOxy^  -  Zy\ 

4.  x^+UxY-llxy^+l5y^^;  5a;5+lla;y+20a:y+9yio. 

5.  ^2a^x  +  13a V  -  x^^;  36a^  -  x^^  +  8a V  -  aV. 

L.  C.  M.  of  Three  Expressions. 

250.  In  finding  the  L.  C.  M.  of  three  or  more  expressions, 
one  finds  first  the  L.  0.  M.  of  the  first  two;  then  it  is 
necessary  to  multiply  by  all  the  factors  of  the  third  ex- 
pression that  are  not  contained  in  the  first  two. 

If  the  three  expressions  are  represented  by  X,  Y,  and  Z, 
and  the  L.  C.  M.  of  X  and  Y  by  m,  we  must  multiply  m 
by  all  the  factors  of  Z  that  are  not  to  be  found  in  m 
already.  To  find  these,  we  divide  Z  by  the  H.  C.  F.  of 
m  and  Z, 

Model  ft. — In  the  three  expressions 

x^  —  y^i  x^  —  y^;  x^  —  y^ 


238  ALGEBRA. 

the  L.  C.  M.  of  the  first  two  is  {x^-y^){x+y) ;  the  H.  C.  F. 
of  this  and  the  third  expression  is  2;^  —  ?/^;  hence  the 
L.  C.  M.  of  all  is  {x^  -  y'){x  +  y){x'  +  y^), 

EXERCISE    CXIX. 

Find  the  L.  C,  M.  of  the  five  sets  of  expressions  in 
Exercise  CVI;  also  of  the  followirig  five : 

1.  4.x^-lx-^\  2x^  -  Ix^  -\-  9;  4.x^  —  l^x'^  +  9:?;  +  9. 

2.  x^-Qx^+llx-^)x^-'^x^-bx+^',  32;*— 13a;3+52a;-48. 

3.  a;3  +  2:^;  -  12 ;  x^  -  x^  -  l^]  x^  -  ^x  -  3. 

4.  4a;3  -  72;  -  3;  1-^x^-  lx^\  bx  -  Ax^  -  6. 

5    ^4  _  3^  _  20;  x^  -  Ax^  +  15^;  25  -  4:X^  +  3x\ 

RearraJige  the  equations  in  each  of  the  following  examples 
so  that  each  shall  consist  of  an  integral  algebraic  expression 
equated  to  zero ;    then  find   the   L,   C,  M,    of  those  ex- 


6.  ^'+^1^^^2_|_i.  x^-x-^^^^^x^-nx',  ?^^^=8-^ 
b  X 

X  —  ^:        ^  x^-Y^      X^-\-6V 

46a;  +  7  =  9:?;^  5  _  2a; 

a;2  =  — ^ 

3a; 

10.  a;^  +  8a;2  +  282;  =  7a;3  +  48;  x^  -  14  =  %x^  -  19a;; 

hx^  +  28  =  17a;2 


CHAPTER  X. 

INDICES;   SURDS;  ROOTS. 

251.  A  Power  is  a  number  whose  factors  are  equal. 

One  of  the  equal  factors  is  called  a  Root. 

The  number  of  equal  factors  is  called  the  Index. 

a^a^  =  aaa,  aaaaa  =  a^ 
Law  I.  a^a^  =  a^  +  ^ 


-         o      aaaaa  „ 

a^  -T-  «^  = =  aa^  a^ 

aaa 


Law  II.         a""  -V-  a^  =  -„  =  oj^*^. 
-  a^ 

When  y  =  X  the  quotient  in  (ii)  becomes  a^; 
When  y  ^  X,  the  quotient  in  (ii)  being  a^  "  ^,  the  index 
a;  —  ^  is  a  minus  number. 

Thus  ^3  _^  ^3  __  ^3  -  3  _.  ^0 


252.  For  these  new  values  of  the  index  the  old  definitions 
will  not  apply;  but  if  we  consider  them  subject  to  laws  (i) 
and  (ii)  we  can  interpret  them  in  terms  of  the  more 
familiar  symbols. 

239 


240  ALGEBRA, 


^O^m^^O  +  m^^m 


Law  III. 

a-^a^  =  a-^^''  =  a^  =  1 

a~V=  1;  a-^~  -g; 

d-m^m  ^  ^-m+m  ^  ^0 

;  a^  = 


Law  IV.  a-'^a'^  =  1:   a-^  =  — ;  ^^^  =  — - 


EXERCISE  CXX. 

State  in  the  simplest  form  the  numerical  value  of  the 
following  expressions : 

1.3-3.     ,.  (ip     3.  (.01)-,    ,._L.     «.__!_. 

6.  356(4)-3.       7.    (16)^(6)-.       8.   ^"(i).       9.  9^3- 

What  is  the  reciprocal  of  each  of  the  following  expres- 
sions ? 

11.   2.       12.    3.       13.   10.       14.   1.  16.   100.       16.    \. 

17.   f.       18.   f       19.   .3.         20.   .001.      21.   2.5.        22.    ^. 


23.    100.1. 

24.    1.1.        25.    33.33.        26.   a. 

a 
27.   -. 

X 

a" 
28.    -3^. 

a  —  h                      ,      „ 
29.  .          30.  X  +  y^, 

c 

253.  From  (iv)  it  follows  that  the  reciprocal  of  any 
power  is  the  same  power  with  the  sign  of  its  index 
changed. 


INDICES;  8UBD8;  BOOTS,  241 


EXERCISE  CXXI. 

State  the  reciprocals  of  the  follotuing  expressions  with- 
out using  the  fractional  form : 

1.  ^^;  ^-^•  \,\  ^3;  ^'"^      2.  \'     3.  {a^+x^)\ 

7.  16-2.       g    2«-^       9.  «-^>.       10.  ic-2  +  3^3^ 

Reversal  of  Signs. 

254.  The  law  of  reversal  of  signs  in  subtraction  has  a 
close  analogy  in  multiplication : 

In  subtraction :  To  use  an  expression  as  a  term  of  the 
SUBTRAHEND  is  the  same  as  to  use  its  negative  as  a  term 

of  the  MINUEND. 

In  division :  To  use  an  expression  as  a  factor  of  the 
DIVISOR  is  the  same  as  to  use  its  reciprocal  as  a  factor 

of  the  DIVIDEND. 

Or,  conversely: 

Any  TERMS  of  an  expression  may  be  replaced  by  a  sub- 
trahend containing  the  negatives  of  those  terms. 

Any  FACTORS  of  an  expression  may  be  replaced  by  a 
DIVISOR  containing  the  reciprocals  of  the  factors. 

255.  Dividend,  numerator,  and  antecedent  are  in  Algebra 
interchangeable  names;  so  also  are  divisor,  denominator, 
and  consequent;  every  example  of  division  being  a  fraction 
or  a  ratio,  according  to  circumstances  which  are  not  always 
apparent. 


242  ALGEBRA. 


EXERCISE    CXXn. 


Transfer  all  x's  and  y's  to  the  divisor,  and  all  a^s,  Vs, 
c's,  and  d^s  to  the  dividend,  wherever  they  occur  in  the 
following  fractions  {or  ratios),  changing  the  index  lohen 
necessary : 

-    x''^  x^  x^y  axy 

'    „-2  •        2.   —5.        3.   ~~ro*7*        4.   ttttq 


a  *  a 


—2\'       4.  ~^.       6.  a-^x^:h^cy-\ 


0?  .V  +  ^  ^"^ 


6-    ^-  7.    7 TTw..  8.   ^^TT--       9. 


(a  +  ^,)-i-  °-      a-3  •     ^-    l  +  a= 


2' 


^"-    a-^  +  52'  ^^-  Z>y2(^2  _  ^3)- 


256.  Model  A.      3a^^>3,o^-^  ^  3jb^^  ^  24g^ 
^~^g~^x^yz~''^         ¥dx^y         l)^dx?y 


EXERCISE  CXXIII. 

Change  to   expressions  that  have  no  zero   or   negative 
indices : 

1.  (j£^x-^        2.  — s.  3.  -^ — y— 5.        4.  a^^cd^e-^-^, 

x~^  x^y~^z~^                           '' 

x?y~^              hx"'^  al)~^                     (2a:^)~3 

^-  ~ar^'       ^   3f^''  ^'  3^^'              ®-  Sx-'y^'' 

2-'^x-Y  (Qay)-' 

9.  TTT-T^-  10.    — ^     ^^ 


Simplify  the  following  expressions : 


y.2    ^j2 


11.  a^"^  +  3/^.         12.  a'^  —  I>-^.        13.  ;::z2 — ^« 


X   ^  — 


y 


x^  —  5^  +  6  a~^  A-  h~ 

14.   —, ^T-V.  15. 


^x-^   '  ■'*'•  a-b  +  h^a'^' 


IJSrmCBS;  sunns ;  MOOTS.  243 

ab-^  +  b{a  4-  b)'^ 
a(a  —  b)     —  a'^b' 

17.  (a-^  -  b-^){a{a  +  b)'^  +  Z>(a  -  b)-^]. 

18.  (a;"^  —  y'^)(j^~^  —  2/"^)^- 

19.  (^  —  y^x~'^){x^  +  y^Y^i^y'^ '-  ^V^  +  ^~V) 

20.  (^y"^  —  yx~^){y^^  —  a;"^)"^. 

When  a  Root  is  itself  a  Power. 

257.  In  4  X  4  X  4  =  64;     2^  x  2^  x  2^  =:  64  there  are 
two  indices  to  consider  .  .   .  and  factors  within  factors. 

(a^Y  =z  aVa^aV  =  aaaaaaaaaaaaaaa  =  a}^. 

Here  for  each  of  the  large  equal  factors  there  are  three 
of  the  smaller  ones;  in  all  five  times  three  smaller  factors. 

Here  for  each  of  the  large  equal  factors  there  are  x  of  the 
smaller  ones;  in  all  Qx  of  the  smaller  factors. 

Law  V.  {a^y  =  a^^. 

268.  Kef  erring  to  (y),  we  have  a  power  a^^ 

of  which  a^  is  a  root 
the  index  being  q. 

This  relation  may  be  written  as  in  (y),  but  also 

Similarly  a^  =   Va^,  since  a^  may  be  arranged  in  three 

equal  factors  thus  :  aa.aa.aa;  and  again  a^  =  Va^"^,  since 
a'^^  may  be  arranged  in  seven  equal  factors,  each  of  which 


244  ALGEBBA. 

contains  x  factors  a.    In  general,  if  we  have  Vc^,  and  if  p  is 
a  multiple  of  q^  we  may  arrange  the  p  factors  of  aP  in  q 

equal  groups;  each  group  will  contain  —  factors  a\  and  in 

that  case,  if  ^  is  a  multiple  of  q. 


EXERCISE    CXXIV. 

Express  as  powers  of  prime  mirribers: 

1.  (2i3)(84)(4326)(162o«).  6.  (8)23  ^(16)1^ 

2.  (92ii)(27i^s)(6ioo)(833).  7.    {(24)12(54)6}  11. 

3.  (12)3(36)^(192)43.  8.   [(1000)«  --  (25)io]i8. 

4.  (200)4(135)38(225)51.  9.   V332 .  9^^ .  81. 

6.  (45)11(675)31(375)28.  lo.   V{VZy\lS')  -  2^. 

Fractional  Indices. 

269.  If  ^  is  not  an  exact  multiple  of  g,  then  the  expres- 
p 
sion  ««  becomes  meaningless,  as  we  cannot  have  a  fractional 

number  of  factors.  We  are  at  liberty  to  assume  that  laws 
(i),  (ii),  and  (v)  apply  to  these  new  indices,  and  under 
that  assumption  to  seek  an  interpretation  for  them. 

a^a^a^  =  «3  +  ^  +  ^  =  a*  =  a^^ 

Here  c?  appears  as  one  of  the  three  equal  factors  of  a^\ 
that  is,  a^  =  VdK 

\a^/   =a^,  ax  —  ya^ 
(a^J   ^aP;  aa  ~Va^ 


mDICES;  SURDS;  BOOTS.  245 

In  general  any  power  whose  index  is  a  fraction  may  be 
taken  as  a  root,  and  as  many  of  them  multiplied  together 
as  the  denominator  indicates;  then  the  product  so  obtained 
will  be  a  power  whose  index  is  the  numerator. 


(J)'  =  a^ 


or,  otherwise, 


Qr 


Law  VI.  UQ  =  Va^ 

260.  By  (vi)  any  root  may  be  exhibited  as  a  power  with 
a  fractional  index. 

261.  Again 

J  =  \Jf  =  ^J^P  as  well  as  a^  =  {a^f  =  V~^. 

A  fractional  index,  then,  indicates  both  a  root  and  a 
power ;  the  denominator  indicates  a  root,  and  the  numer- 
ator indicates  a  power ;  and  either  the  root  or  the  power 
may  be  found  first. 


EXERCISE    CXXV. 

Express  with  radical  signs  : 

1.  «'.  2.  x^.  3.  {a  —  Z>)i      4.  a^b^. 

6.  a^M.  6.  x^2/'t  7.  x^y^  (compare  Ex.  6). 

8.  p^q^  (compare  Ex.  5).        9.  a^xh  10.   (ax^y^)^, 

Express  tuith  fractional  exponents  : 

11.  Va^.  12.  ^^a^.  13.  i^x^a^.         14.  Vax^y^. 

15.  \^{a-hf.    16.  V^-b^      17.  t^Sy.         18.  'f^. 

19.  ^\/a^xh^y^,    20.   ^ . 

yxy^ 


246  ALOEBBA. 

Give  the  numerical  value  of  each : 

21.  8?.  22.  165.        23.  32S.  24.  (i)?.        25.  (|J)*. 

26.  (.36)5.  27.  (.008)i       28.  (1.728)1.     29.  -(15625)i 

30.  (256) -8.      31.(^^7)-?.      32.  (-8)-?.      33.  (64)-Ki)- '. 
34.  (.0001)-J^         35.  (81)-5(.3)l  36.  (i)-'(243)-S. 

37.  {i^)-^{Vll)-K  38.  (216)i(T-V)-^(.l)-^. 

89.  (.0625)-3(i)-=.  40.  l(128)5(V^)-?32-«6-2}-i. 

41.  4-^  42.  92».  43.  16".       44,  32«.         45.  256>2'. 

46.  iiY'.  4     (H)'-^.       48.  (W)"'-     49.  (2.25)^^ 

Simplify : 

61.  (aZ^^)i  52.  (a^Z^-^)^  53.  (tt^Z^-^)^. 

54.  {a^^)h  56.  (Vj-^)-i  56.  (8a;y;2«)*. 

67.    {xVyf.  58.  (6?2|/^6^  59.    (  VZQx^y. 

^O.Va^Va^.  61.  (V(«+a;)^^.  62.  \^x\x-\-y)^, 

RADICALS. 

262.  A  radical  expression  is  one  that  contains  a  radical 
sign;  a  rational  expression  is  one  that  can  be  written 
without  a  radical  sign  or  a  fractional  exponent. 

263.  A  surd  is  a  radical  expression  that  cannot  be  shown 
to  be  rational. 

264.  A  surd  whose  index  is  2  is  called  a  quadratic  surd; 
if  the  index  is  3,  a  cubic  surd;  and  if  necessary  similar 
names  could  be  found  for  radicals  of  index  greater  than  3. 
Equiradical  surds  are  those  having  the  same  index. 

266.  A  mixed  surd  is  a  surd  with  a  coefficient  outside 
the  radical  sign  ;  an  entire  surd  is  one  without  such  a 
coefficient. 


INDICES;  SUBD8;  ROOTS.  247 


EXERCISE   CXXVI. 

Express  as  entire  surds  : 

1.  2  4/53.          2.  3  Vl.           3.  18  V2. 

4.   .01  4/1000. 

5.  29  V%.          6.  2  V¥^.          7.   Ill  V'lT. 

8.  3  i/87. 

9.  87  fS.       10.  100  1^.00089.       ii.  3  ^9. 

12.   Gf'S. 

IS.   11  V'lll.     14.  2  ^.           16.  4  f  2, 

16.  2  f  3. 

17.   3  ^2.            18.   17  f3.        19.  17  4/3. 

20.  2f3. 

21.  3  f  2.            22.   2  f  27.         23.  3  f  16. 

24.  13  VTi. 

26.    VI  H.        26.  |/27f3.     27.   V'^  V^. 

28.   ^2  Vs. 

29.    fS  1/3.        30.    VI  VI. 

266.  A  surd  is  sometimes  called  an  irrational  expression. 
Similar  surds  are  those  whose  irrational  factors  are  the 
same. 

267.  A  surd  term  is  said  to  be  in  its  simplest  form  when 
it  is  written  with  only  one  radical  sign ; 

no  rational  factors  in  the  radical ; 

no  fractions  in  the  radical ; 

no  radical  sign  in  the  denominator. 
And  in  general  any  algebraic  expression,  to  be  in  its 
simplest  form,  must  have  no  fractional  or  negative  ex- 
ponents. 

EXERCISE  CXXVII. 

Express  in  the  simplest  form  : 

1.  /i2.      2.  v'Is.      8.  V50.      4.  V108'.      6.  VaooT 

6.    1^448^      7.   V'm.       8.   1^1584.     9.    1^1728.     lo.    ^"845^ 

11.  .0024/500000.  12.  WVTQ,  13.  ^^^^^ 

58     • 


248  ALGEBRA. 

14.  .004^6250000.  i8.  ( ~^^        16.  13-^(l''3187)-*. 

17.  V^  -^  (V«'-9a2+27a-37)"'.     18.  4«'i5-  ^I^S^sFl 
19.  275:r^:  Wlb^^f.     20.  15a2(^2_^2)-3.  3^-2|/(^_^)-3^ 

Z77iiYe  similar  surds  in  the  following  expressions : 
21.  VS-  2VE0+  V98?  22.  4/343"-  ^27"-  4/12". 

23.  1/507+34/12-8 1/75+ 1^507    24.  4/1000  +  4/360'-  4/207 
25.  v'800  -  4/8OO  +  4/270  -  V72". 

26. 4/32  +  I/32  +  4/108  -  |/ia: 

27.  4/48  -  4/8I  -  4/675  +  V'648T 

28.  >^320  +  I/5OO  -  VT372  +  1/2560. 

29.  1^6400+4/1445-4/405".  30.  4/29I6+ 1/432-1/250^ 

Distributive  Law  for  Indices. 

268.  The  expression  a'^V  contains  7  factors  each  of 
a,  h,  and  c\  with  one  of  each  in  a  set,  there  would  be  7 
sets  of  factors, — 

a'Z>V  =  {ahc)\ 

In  general  powers  of  different  roots,  having  the  same 
index,  follow  this  law :  The  product  of  the  powers  is  equal 
to  the  power  of  the  product. 

Law  VII.  a'^h^'x''  =  {abx)\ 

269,  This  law  holds  also  for  quotients : 

a'^P       db    ah      f  abV 


_  db    ab  __  lab y 
—    c   '  c  ~  \  c  I 


INDICES;  SURDS;  ROOTS.  249 

270.  The  same  law  holds  for  roots: 

Va  VI  =Vah;  because  Va  VI  V~a  Vb  =  Va  Va'Vb  Vb  =  ab. 

271.  Similarly  for  all  fractional  exponents: 

a'^w)  ~  (aby 

p  p  p 

.-.     aW={aby 

Hence  the  law  (vii)  holds  for  all  exponents,  positive 
or  negative,  integral  or  fractional. 

272.  Kadicals  with  different  indices  can  be  reduced  to 
radicals  with  the  same  index  by  expressing  them  with 
fractional  exponents  and  reducing  the  fractional  exponents 
to  a  common  denominator.  Thus  any  two  radicals  can  have 
their  product  or  quotient  written  as  one  radical. 

Model  B. 

EXERCISE    CXXVIII. 

Reduce : 
1.    ^2^2.         2.   VSf'IO.         3.    1^6^12.         4.    V6V2. 
6.    V2V6.  6.    V5V3,  7.    V3V5.  8.    VbVI. 

9.  VibVI.      10.  VqVI,     11.  V^V^V^, 
12.  VsVdVd.     13.  V2V3VI.     14.  V2Vf, 

16.   V2VJ.       16.   VavW,       17.    VaVb.       18.   VaV^. 
19.  VaV¥.      20.   V^  V^b. 


250  ALGEBRA, 


Rationalizing  a  Term. 

273.  Any  radical  term  may  be  converted  into  a  rational 
term  by  multiplying  it  by  a  suitable  factor.  This  operation, 
however,  changes  the  value  or  the  term. 

V^V2  =2 

f  2  f 4  =  f  8  =  2 

If  the  radical  term  is  in  the  denominator  or  in  the  nu- 
merator of  a  fraction,  we  may  take  advantage  of  this  opera- 
tion WITHOUT   CHANGING   THE   YALUE   OF   THE   FRACTION. 

Usually  it  is  desirable  to  have  the  denominator  rational 
rather  than  the  numerator.     Thus  —-=  may  be  written 

V^V^       2  V^VY      Vi4 

or    — r=^ 


but  the  second  form  is  easier  to  calculate,  7  being  a  simpler 
divisor  than   Vl4:, 

EXERCISE    CXXIX. 

Rationalize  divisors  : 


_i_  _1_  J^  |i  i- 


2.  -^.      3.  —'       4.  JJ.      6.  -3^. 

i^I  V3  3  3  1135 

11.   -^:^.      12.    — —.      13.   — —*      14.   — -7=.      15.     " 


3  11^ 
nJ  32 


3f3  '  2V2  '  4V2  *  2^2  '   N  32 


d                a^                  a              Vxy 
16.    — ;=!.      17.    -7=.      18.   ^  h-.      19.    7_. 

I^^»  VaJ  N^  yVic 


yVx         '    V^3a 


INDICES;  SURDS;  BOOTS.  251 

af8?  ZaV^  a^hcVoFc  \^32a^ 

|/i25  f  9^ 


6.  --=:^.     26.      , ..     27.    Vl5w  :  f225a;l 

1/5;^;  4/3^^2 


28. 


5i/50a2&  :  2VlOab,     29.    'f/l25«-«^>- 


30.  VlOOOa^-^  :  200a&. 
C7>2iYe  similar  surds  in  the  following  expressions : 

31.  V^-^l  +  ^l-^l' 

'^-  \|i8  +  \|50~\l50  +  \27' 

'*•  nJ32  +  \|18  ~  sjas  +  Nir 
'®-  \|i6  ~  \|¥  +  sJm  ~  "^s/so" 

\4«      \|3a;^\|3"  \J108a;  ^  \|l00a;2      Sj: 

39.    V.03.  +  ^->^-^. 
^^'  \|     5      +nJ5Z>2^~\20c3* 


5x 
32* 


262  ALGEBRA. 


Rationalizing  Quadratic  Surds. 

274.  Two  binomial  quadratic  surds  which  consist  of  the 
same  two  terms  with  opposite  signs  between  them  are 
sometimes  called  conjugate  surds.  Their  product  is,  by 
Theorem  A,  a  rational  expression.  Where  one  such  factor 
occurs  as  the  denominator  of  a  fraction  we  may  utilize  this 
fact  to  rationalize  the  denominator. 

Model  C. 
Model  D. 


5  +  1/5  25-5  20 

One  important  advantage  of  such  reductions  is  evident 
when  we  seek  the  value  of  the  expressions,  having  given 
the  numerical  value  of  the  square  roots.     Thus 

^=2.236;     ^^=^,;     ^)5-  ^5)=?-(2.764); 
the  reduced  expression  is  obviously  easier  to  calculate. 

EXERCISE    CXXX. 

Rationalize  the  denomiiiators : 

2+V2 

2  -  V^'        "'  3  -  W      "  2-  V2 

3  +  21/5' 


5.  -T= 


V2- 

1 

v%  + 

1 

4/2- 

1 

3  + 

V2" 

3-51^5 

9    8  +  4  V3" 


3. 

7 

2 

3-4/3"          ■ 

-1-4^5" 

10. 

1-4/2"  ' 

10  V2  +  2  4/5" 

g        ^     ' _.    9.  ■  _.    10.  '       _ 

•  - 1  +  |/2  6  -  2  V3  5  -  2  i^lO 


INDIGES;  SUBDS;  ROOTS,  253 

Find  the  values  of  the  expressions  in  the  preceding  exer- 
cise, assuming  the  folloiving  approximate  values  for  the 
roots  : 

1/2"=  1.414;    V3  =  1.732;    4/5"=  2.236;    |/lO  =  3.1623. 


275.  Two  sncli  multiplications  are  necessary  to  rational- 
ize denominators  which  are  quadratic  surds  of  more  than 
two  terms. 

EXERCISE  CXXXI. 

Rationalize  denominators : 

3  +  1/6^  +  2^2  |/6-_  |/io  +  4/15     -         3  +  V^ 


4. 


3  -  l/6a  +  2^2          i^6~+  VlO  -  VTb         1+  l/2  +  Vs" 
2-1/5"  1 


Other  Rationalizing  Factors. 

276.  Rationalizing  factors  can  also  be  obtained  for  bi- 
nomial surds  that  are  not  quadratic ;  but  their  complexity 
often  makes  it  undesirable  to  seek  for  them. 

Model  E.      V'^  —  Vo  suggests 

a^  -  b^^(a  -  b)(a^  +  ah  +  P) ;     so  that 

(|/2  -  H)(H  +  ^10  +  |/25)  :=  2  -  5  =  ^  3. 

Model  F.     2  +  i^3y    suggests 

{a  +  b)(a^  -  a^  -  aP  +  b^)  =  a'  -  b';  so  that 

(2  +  f^3y)(8-  4  t'S^  -  2  ^9f  +  ^27p)  =  16  +  3y. 


264  ALGEBRA, 

EXERCISE    CXXXII. 

Find  rationalizing  factors  for : 

1.  1-^2.         2,  2  +  H  3.    1^2""-  H 

4.    t^2  -  1.         6.     V2  -  V'S'. 

IMAGINARIES. 

277.  Imaginary  quantities  present  an  important  excep- 
tion to  the  algebra  of  ordinary  radicals,  in  this  respect  : 

V  —  2  must  by  definition  give  —  2  when  multiplied  by 
itself;  law  (vii)  can  therefore  not  apply  to  imaginaries, 
because  by  its  application  we  should  have 

(-  2)^(~  2y  =  [(-  2)(-  2)]*  =  (4)*  =  VI. 

This  would  allow  V—  2  to  be  equal  to  4^2. 

278.  If  we  allow  i'^  co  stand  for  —  1  and  i  for  V—  1,  as 
in  a  previous  chapter,  we  shall  have  V  —  a  =^  Vi^a  =  i  Va\ 

then     (  V—  a){  V—  a)  =  {i  Va){i  Va)  =  i^a  —  —  a. 

The  symbol  i  can  be  handled  according  to  the  ordinary 
laws  of  algebra  and  subject  to  the  definition 

%'  =:  -  1. 

It  follows  from  this  that  i^  =  —  t;  t^  =  1 ;  and  in  the 
expression  i'^  we  may  subtract  from  n  any  multiple  of  4 
without  altering  its  value. 

279.  An  expression  involving  the  square  root  of  a  nega- 
tive number  is  called  an  imaginary  (or  a  complex)  number; 
and  i  is  called  the  imaginary  unit.  All  imaginary  expres- 
sions may  be  reduced  to  the  algebra  of  ordinary  radicals  by 
substituting  i  ■—  V  —  1  as  a  coefficient  of  the  radical;  and 
of  course  no  term  containing  i  can  be  regarded  as  similar 
to  a  term  not  containing  i. 


INDICES;  SURDS;  BOOTS.  255 

EXERCISE  CXXXIII. 

Unite  similar  surds  : 


1.   V^^  +  V^^^^^  -  i/^=~T8". 

3.  V-  72  -  V-  243  +  f'-  338  -  V-  1200. 

4.  3  V^^  +  8  /^^  -  l/6"(  |/-  50  -  V-  12). 

5.  3  l/-  32  -  2  f/^^^^"^  -  V^^nJ{  V^+  2). 

/ory  

6.  V-  200  -    3Jy  +  v^Kt'"^^^^^  -  '^135). 

<J-4-f) 


7.    4/_  200  -  '^/3(  J4^  -  J-  q^  )  +  1/200. 

I-  4 


8.  i/-  500 

9.  V'^^TS  -  f/^^^^  +  |/48  -  V-  108. 

10.    4/1250  -  V-  50  +  1^150  -  V-  450  -  |/24. 

Simplify: 


11.   ( |/2  +  |/-  1)2.  12.  (  |/2  +  V-  2)3. 

13.  ( i/2  +  v^^y,        14.  ( 4/5  +  v^^y. 


15    (21/3-3  |/=^)2.        16.   (  V-  12  H-  V-  18)3 

17.  (  |/50  +  4/'^=^27)(  4/48  -  4/^=^). 

18.  (  4/8  +  4/^=^  -  4/200)(  4/-  32  -  4/^=^"675). 


266  ALGEBRA. 

Square  Root  of  a  Binomial  Quadratic  Surd. 

280.  Prom  the  identities 

{Vx  +  Vyf  =  x  -\-  y  +  %  Vxy 
{Vx  —  V^y  =  x-\-y—2  Vxy 

we  conclude  : 

The  square  of  a  binomial  quadratic  surd  is  a  binomial 
surd  with  one  term  rational. 

And  also  : 

The  square  root  of  a  binomial  quadratic  surd  of  which 
one  term  is  rational  is  the  sum  (or  the  difference)  of  the 
square  roots  of  two  numbers  such  that: 
their  sum  equals  the  rational  term; 
their  product  equals  the  square  of  half  the  surd  term ; 
in  the  given  binomial  surd. 

EXERCISE   CXXXIV. 

Find  the  square  root  of : 

1.  3  +  2  V2.  8.  35  +  1^  ^.  15.  1338  -  72  V^. 

2.  7  -  4  1^  9.  23  -  8  |/7.  16.  3.31  -  .22  V210, 

3.  5  ~  2  i/6.  10.  310  -  66  V2L  17.  12.55  -  .7  Vm, 

4.  6  -  2  1/5.  11.  9^-3  V2.  18.  71  -  -V-  V2T. 

6. 7  +  4  4^;    12.  H  -  v^.      19.  w-  -  f  ^^• 

6.  19  -  6  |/10.   13.  lOyV  -  i  ^30.   20.  64.8  -  1.6  VSO. 

7.  33  -  20  V2.    14.  70  +  40  Vd. 

281.  When  both  terms  of  the  given  expression  are  surd, 
the  square  root  cannot  be  found  without  removing  a  surd 
factor,  the  square  root  of  which  must  be  found  separately. 


INDICES;  SURDS;  BOOTS.  257 

EXERCISE  CXXXV. 

Find  the  square  root  of: 

1.  16  4/3  -  2  Vl65.     3.  5  Vy  -  7  i^.     6.     7  |/2 -4  4/5^ 

2.  13  1/7  +  2  1/2I0.     4.  8  V2"+  2  4/I4. 

EXERCISE    CXXXVI. 

Substitute,  in  the  following  expressions, 

x=  Vq;  y  =  hVq';  z  =3  V^k; 
and  simplify  the  results : 

o^c  x^y  ^  ^  27 

6.  45ay  -  120ay.      7.  hbx^Y  -  l^^xhY^       8.   ^^^f^. 

X  —  z 

^    h"  -x^  ^^      x'-z"^ 

y'^  —  a^'  '  {c^  —  y'^y 

In   the  same  expressions   substitute   (and  simplify   as 
before)  : 

a=VI;  &  =  5  f 8;  ^  =  ^1^5'; 

x  =  q^;  y  =  hq-i',  z  =  30l'-'M. 

In   the  same   expressions  substitute   (and  simplify   as 
before)  : 

9a  2b  , 

6a  b  , . 

7bVb  ^      V'da' 


268  ALGBBBA. 

Square  Roots  of  Complex  Binomials. 

282.  In  finding  square  roots  of  binomial  surds  we  often 
meet  imaginary  expressions. 

EXERCISE    CXXXVII. 

Find  the  square  root  of: 
1    2i/"^-l.       4.6-6  4/"^=^         7.-4-6  4/^=^5: 


2   1  +  2  |/  -  2.       5.  26  +  4  4/ -30.     8.  -  19  -  10  V-  6. 


3.  2  +  4  4/  -  2.       6.  118  +  2  V  -  3.     9.  51  +  14  4^  -  10. 


10.   8  +  24/-  65. 

Radical  Equations. 

283.  Some  radical  equations  can  be  solved  by  taking  the 
radical  itself  as  the  unknown  letter  (and  in  that  case  the 
problem  is  much  simplified  by  letting  z  stand  for  the 
radical). 

Model  G. 
^  1^  _  2  Vx-  5  _  3  Vx-  7 
^        Vx  +  b  ^     2  Vx 

Let  z  —  Vx'. 
7      2z-5  _Sz-'r 
®4       z  +  5  ~      2z 

(D  7z^  +  35^  -  8^2  _|_  20z  z=  Qz^  +  I62;  -  70      ©  X  4:z{z+  5) 
®  7;^^  -  39;^  -  70  =  0  (D  +  ^^  -  55;2 

®  (7z  +  10)(;2  -  7)  =  0  ©  factored 

Whence z=  — ^;  z  =  7. 

But  z  =  Vx:  hence  x  =  -77,- ;  a;  =  49. 

49 


INDICES;  SURDS;  BOOTS.  259 

Notice  that  x  =  -j^  will  not  satisfy  the  given  equation 
unless  the  radical  is  taken  with  a  minus  sign. 
Model  H. 


®   V3X  +  4:  +  V5x  +  1  +  i/lSa;  -5  =  0 
©  Vdx  +  4  +  V6x  +  1  =  -  VTSx~^        ©  -  Vl8x^^6 
(D  3X+4.+  2  V{5x+l){dx-\-4:)+6x+l=lSx-5     (zf 
(A)  2  V{iDX^  +  23x  +  ^)  =  lOo;  -  10      ©  -  8a;  -  5 
®  i^nx^-{-23x  +  4.  =  5x-5  ®  -^  2 

®  15x^+2dx+4:  =  26x^-50x+25'        ®2 
®  10a;2  -  73a;  +  21  =  0  ®-15a;2-12a;-4 

®  (a;  -  7)(10a;  -  3)  ==  0  ©  factored 

Whence  a;  =  7 ;  a;  =  3^. 

Notice  that  ®  if  squared  would  give  three  double  cross 
products;  ©  squared  gives  only  one;  and  a  similar  reason 
holds  for  the  change  from  @  to  ®.  Again,  we  must  notice 
that  the  value  a;  =  7  satisfies  ®  only  if  we  take  the  sign  of 

1^1 8a;  —  5  minus;  and  the  value  a;  =  .3  satisfies  ©  only  if 

l^5a;  +  1  and  i^l8a;  —  5  are  both  minus. 

EXERCISE    CXXXVIII. 

Solve  the  equations : 

4i^-l        .^-.    3a;  +  9 
1.  8  r  a;  H ^ir-  =  5  rx -{ —=. 

%-Vx  1  +  Vx 

Vx  +  4:      i^-4      24  3Vx  +  2      ^d  +  2Vx' 

'  Vx-4:      Vx  +  4:       5*         '  1  +  Vx  6  +  2Vx' 

4:Vx  b  -2Vx      21 

4    __  —  _ L    . 

*  10  +  Vo:       2  Vx  +  4:       8 


260  ALGEBRA. 

Vx  +  3  ___  4  _   3  4^^+1 
i/x-2       3  ■"  2(1  +  4/^)' 

6.  V^x-  3  +  Vx  +  '^  =  V9x  +  1. 

7.  4/4^1  +  Vx  -4  =:  |/8^. 

8.  Vila;  +  11  +  Vx^^  =  4/20:c  -  4. 

9.  V2x  +  3  +  l^i?;  -  2  =  4/52;  +  1. 

_  2 

10.    4^40;  —  S+Vx-2  =  Vx.      11.    4^:^;  +  1  +  V^  =  - .— 

4^ic 

/»  q 

12.    4/2; +  4  +  4/^  =  -—  13.     V4:X  -  Q  +  Vx  =  -— . 

4^  4^iz; 

14.    VQx~+~^  +  Vx=-^.     15.    4^17:?; +  28  +  4^^=-^. 
4^a;  Vx 

16.    4/.^  -  2  =  J4^.       17.    Vx^^  =  V5x~+1. -- V2x+~3. 

18.  4/^?;  —  2  =  4^9^  -  2  —  4/50;  +  1. 

19.  34/^  —  1  —  4^4^;  +  9  =  Vx  —  5. 

20.  4^7S~+T  —  4^20;  +  9  =  Vx  +  5. 

6  2         ^  3,1       24^+  1 

21.    ~= y==  3.  22.  — =  +  -^=  '■ . 

1  —  Vx        Vx  Vx       ^  ^ 

23.  V2x  —  Vx  —  34  =  Vx  —  62. 

24.  Vx  +  1  +  i  Vx  -  4:  =  V2x, 

2  __        3 

25.  1/15:?;  -  5  +  Vx  =  ^-     26.  2  4^^  -  8  +  l^i^;  =  — ^• 

27.  4^52;  +  1  +  Vx  —  2  =  4/90;  —  2. 

28.  4^9^  —  9  —  V4:X  +  9  =  4/.'?;  —  5. 

29.  4/9^"+~l  —  4^3:r  +  3  =  Vx  +  5. 

30.  V7F+~i  -  V2x  +  9  =  4^a;  +  5. 

31    .  p-^^^—  +  1  =  -.  32.  4/a;  +  5  =  ■— =  -  Vx. 

\       X  '  X  ^  ^x 


mnicm;  suhds;  moots.  261 

4 


83.   Vx  +  6  +  Vx-^1  = 


34. 


35. 


36. 


Vx-1 
1  1  3 


2  +  Vx       2Vx  -  4        10 
2  +  V^       1-  Vx  _  9_ 
2  --  Vx        1  +  Vx'~  ^' 
11  3 


Vx        1  +  Vx        10(2  +  Vx)  ' 

38.  t/^qri+J^=  ^Vs^^TTT. 


39.    i^Sic  +  1  +  |/5.T  +  4  +  i^a;  -  19  =  0. 


40.    |/2a;  —  Vx  -  '34:  =.  Vx  -  62. 


41.    1^82;  +  1  -  V6x  +  4:  =  Vx  -  19. 
3  1^^-  2       2  4^-  3       15 


42. 


2Vx-  d       dVx-  2        4 ' 


r  ./~       15  —  Vx 
43.  5  r o; =r  =  15. 

4:  +   Vx 


44.  y4a;  +  3  -  |/.T  +  2  -  1^90;  +  1  =  0. 

45  «,.        15  Vx 

45.  7=  =  75 f^— . 

5  -  Vrc  7 


46.  Vx  +  2  +  |/a;-  1  = 

47.  Vx  +  6  +  Vx  = 


Vx  +  2 
3 


i^cc  +  5 
5 


48.    i/o;  +  8  +  1  =  - 

Vx^  +  8a; 


262  ALGEBRA. 

11 


49.    |/2a;  -  79  =  i^  + 


50. 


V'Zx  -  79 
13  2 


\/x-\-2        5        1  -2V\ 


X 


\x  +  20      a;  +  20 


Equations  Solved    Like   Quadratics, 

284.  In  the  following  examples  the  expression  under  the 
radical  sign,  or  some  multiple  of  it,  can  be  put  together 
out  of  the  terms  that  form  the  rest  of  the  equation,  in  such 
a  way  that  if  z  be  substituted  for  the  radical  the  equation 
may  be  treated  as  a  quadratic  in  z.  After  solving  this  the 
radical  is  again  substituted  for  z,  and  that  equation  solved 
for  X, 

Since  the  given  equation  would  be  of  the  fourth  degree  if 
cleared  of  radicals,  four  values  of  x  are  expected. 

Model  I. 

1  -  \/2x^  ~-3a;  -  5  _  3  -  2a; 
^  X  ~        9 

©  9  —  9  V^x^  —  ^x  —  b  =  3x  —  2x^  (T)  X  9x 

(D  2x^-dx+9-  9  V2x^  ^3x-6  =  0  (2)  -dx+2x^ 

®  2:c2-3a;-5-9  V2x^-3x-6+14:=0  Same  as  (3) 
®  z^-9z+14:=0           [in  this  equation  z  =V2x^-3x--5'] 

®  {z  -  7){z  -  2)  =  0  ®  factored 

(T}z-'7  =  0;z-2  =  0  ®  Ax.  A 

(B)z  =  7;  z  =  2  from  © 

®  1/2^21:3^35  =  7;  2x^-dx-5=4:d  from  ® 

@2x:^  —  dx  —  64:  =z  0  ®  -  49 


INDICES;  SURDS;  BOOTS,  263 

(0)  (2a;  +  9) (a;  -  6)  =  0  ®  factored 

9 

@2x^  -  Zx  -  b  =  ^  from  ® 

@  2.^2  -  3a;  -  9  =  0  @  --  4 

(15)  (2^  +  3)(ii;  -  3)  =  0  (g)  factored 

3 
(iDa;=--;a;=:3 

9  3 

Ans,  ic  =  3;  a;  =  6;  a;  =  —  — ;  a;  =  —  -. 


EXERCISE  CXXXIX. 

In  the  same  way  solve : 


1.  4a;2  -  13ir  =  ISV^x^  -  7^?;  -  28  +  ir. 


2.  :r  +  2a;2  +  sVx^  -  3:?:  -  9  =  7a;  +  12. 


8.  x^  —  4:X  —  MVx^  —  5a;  —  2  =  a;  —  62. 

a;  -  7       Vx^  -  4a;  ~  20  -  1 


4. 


a;  +  3 


5.  4  +  V^x^  -  12a;  -  71  ==  ^^{x'  -  4a;  +  7). 


6.  8a;'^  +  a;  -  11  +  VSx^  +  a;  -  5  =  0. 

7.  3a;2  -  ll|/3a;2  __  ^  _j_  23  =  2;  -  53. 


8.   8a;2  +  6a;  -  5(1  +  V^x^  +  3a;  -  1)  =  0. 


9    6(a;2  +  2a;  +  8)  =  l^V^x^  +  6a;  +  19  -  10. 
10.   V7a;''^  —  a;  +  30  =^  |(7a;^  -  a;  +  10). 


264  ALGEBRA, 

EXERCISE  CXL. 

In  order  to  find  the  value  of  x  i7i  the  follotoing  equatio7is, 
it  may  he  necessary  to  extract  the  square  root  of  a  binomial 
surd : 


1. 

x^  -  10a;2  +  1=0. 
x''  -  "^^x^  +  25  =  0. 
^4  __  60^2  _j_  36  ^  0. 

a;*  +  1  =   14:XK 

38a:2  _  ^4    ^  215. 

6. 

7. 

8. 
9. 
10. 

1  =  Qx^  -  x\ 

^2 

2. 

3. 
4. 
5. 

I  =  V5x^  +  7. 

"^^  +  ^  -  28 

x>  -  lb  =  2xVZ. 

4 
The  following  equations  yield  surd  answers,  which  must 


he  reduced  to  their  lowest  terms 
11.  a;2  -  30a;  +  9  =  0. 


X 


21.  —=  Vx  —  4:, 
6 

22.  |-  +  10  =  6(a;  -  6). 

23.  |-  -  10  =  6(a;  +  6). 

24.  (3a;  ~  8)2  =  48a;. 

25.  a;  =  8  +  i|/44a;  +  3. 

26.  7a;2 :  (35a;  -  2)  =  8  :  7. 

27.  16(a;2  +  4)  =  120a;  +  1. 

28.  a;2  + 4.8a; +  1.84  =  0. 
_                               29.  a;2  +  11  =  17.5a;. 

11  —       -  ,,1        .0625 

30.     .16   =: 2— • 

X  x^ 

Express  the  answers  to  the  last  20  examples  decimally, 
assuming 

V2  =  1.414;      V3'=  1.732; 

V6'=  2.236;      VW  =  2.4495. 


12. 

a;2  +  10  =  16a;. 

13. 

a;2  +  20a;  =  50. 

14. 

x^  =  17  -  32a;. 

15. 

x^  +  28a;  +  71  =  0. 

16. 

61  =  26a;  -  x\ 

X^  —  Q? 

17. 

**"         "^    —  \r    \     ^ 

5  —  5a; 

18. 

a;2  +  32a;  +  106  =  0. 

«    .    ^^ 

19. 

-^  =  ^  +  18- 

20. 

"'+'=2.  ^. 

INDICES;  8UBD8;  BOOTS. 


265 


ROOTS. 
Square  Roots  of  Numbers. 

285.  In  finding  the  square  root  of  383161,  we  first  see 
that  the  result  is  larger  than  600  (since  383161  >  360000) 
and  subtract  the  square  of  that ;  then,  concluding  that  the 
remainder  of  the  square  root  is  not  less  than  10,  we  sub- 

383161  619      ^^^^^  ^^^  ^^^^  ^^  '^^^  SQUAKE  of  600  +  10. 
36  Eepresenting  600  by  a  and  10  by  h,  we  see 

that    we    have     subtracted    first    a^,    then 
{2a  +  l)h,  in  all  {a  +  If  or  6101    Next  we 
decide  that  the  remainder  of  the  result  is  not 
11061  1229    ^^^^  than  9,  and  so  subtract  the  rest  of  the 
11061         9     SQUARE  of  610  +  9. 

If  this  process  is  not  concluded  by  a  zero 
remainder,  it  can  be  continued  indefinitely. 

Square  Roots  of  Algebraic  Expressions. 

286.  The  square  roots  of  algebraic  polynomials  are 
found  in  the  same  way. 

Model  J. 


231 
121 


0 


121 

1 


+  Qa^b  -  a«62  -  30a6»  +  25M 


a*  +  Zdb  -  56« 


+  ^a^b  -  a'b''  -  30a6»  +  255-1 
+  6a»5  +  9fl^'62 


An%, 


2a^  +  Sab 
-f  Sab 

2a2  4-  Qab  - 

-562 
-  562 

-  10a''b'-S0ab^-j-26b^ 
-10a'b^-S0ab^-\-2^b^ 

'^       0 

A  large  part  of  the  work  of  this  example  consists  in  re- 
writing the  same  terms,  as  remainders  in  successive  sub- 
tractions.    The  work  may  be  contracted  as  follows : 

Model  J. 

^+J^^^-  aW  -^Qtt^ -\-^M^\d^  +  ^ah  -  5b^ 


-IQt^ 


2^2  -f  Sah 


2^2  ^  Qah  -  5&2 


266 


ALGEBRA, 


EXERCISE  CXLI. 

Find  the  square  roots  of: 

1.  9x^  +  A2x^  +  37^2  _  28:^  +  4. 

2.  4:X^  —  4:X^  +  4:X^  +  x^  -2x  +  1. 

3.  20x^  -  26a:2  -  122;  +  9  +  25a;^ 

4.  6x^  +  26^3  -  11a;*  +  9x^  +  Ux^  -  20a;  +  25. 
6.  49a;*  +  56x^  -  2ix^  -  ^2x^  +  9  +  lQx\ 


Series  for  Square  Root. 

287.  With  algebraic  expressions,  as  well  as  with  num- 
bers, if  there  is  no  exact  square  root  the  process  may  be 
continued  indefinitely.  This  leads  to  a  succession  of 
algebraic  terms  which  will  always  be  incomplete,  however 
far  extended. 

Model  K. 


1  -a; 

1 

X     x^     x^     5a;* 
2      4     16     128 

X 

~2 

x^ 
4 

X^         X^         X* 

""  4  +  8"  +  64 

.  8 

x^      x' 
8  ~64 

^3          x^            X^             X^ 

8  +  16  +  128  +  256 

x'       X' 

a? 
16 

5a;*        x^         x^ 

64       128       256 

5a;*       5a;5       6x'        6x'         25x^ 

64  +  128  +  512   '    1024   '   16384 

x^      afl       5x* 

^x* 
138 

INDICES;  8UMD8;  BOOTS.  267 

EXERCISE  CXLII. 

Find  series  for : 


1.  VT+Jx.  4.    V26  +  X, 

2.  Vr+5x.  5.    V9  -f  Sx. 

3.  Vl  +  ^x-  3a;2.  6.    |/a;2  +  1. 


Cube  Roots  of  Numbers. 

288.  The  process  for  cube  root  is  closely  analogous  to 
that  for  square  root,  and  is  suggested  by  the  identity 

{a  +  hf  =  a^  +  Za?h  +  ^ah^  +  l\ 

Model  L.      1/584277056. 

First  we  decide  that  the  cube  root  is  not  less  than  800; 
and  subtract  the  cube  of  that : 


584  277  056 
512  000  000 


WO  =  a 


72  277  056 

Call  800  a  and  the  next  part  of  the  root  that  we 
expect  to  find  I,  Since  we  have  subtracted  a^,  the  rest  of 
the  cube  of  (a  +  l)  is 

'da?!)  +  ^ah^  +  Z>3  =  j(3a2  4-  ^al  +  V^)\ 

and  the  remainder,  72,277,056,  will  be  larger  than  that. 

Now  if  we  knew  the  value  of  ^a^  +  3a^  +  V^  we  could 
find  h  at  once  by  dividing.  Although  we  do  not  know  it, 
we  are  helped  in  guessing  at  h  by  the  fact  that  Za?  is  much 
the  larger  part  of  the  divisor. 

In  this  case  da?  =  1920000,  and  we  conclude  that  h  is 


268 


ALGEBRA, 


not  less   than  30;    on  the  supposition  that  a  =  800  and 
J  =  30  we  construct  5(3a^  +  ^^^  +  ^^)  ^'^^  subtract. 


584  277  056 
612  000  000 


72  277  056 


59  787  000 


SOO  =  a 
30=  b 


19200^6>  =  3^2 
nOOO  =  3aZ> 
dOO  =  W' 


199296>(9  =  3a2  +  3a^>  + 


12  490  056 

We  have  now  subtracted  the  cube  of  830;  we  may  call 
830  a,  construct  3a^  and  proceed  as  before,  until  there 
is  no  remainder,  or  until  enough  figures  of  the  root  are 
obtained. 


Model  L. 

584  277  056 


512  000  000 


72  277  056 


59  787  000 


12  490  056 


12  490  056 


WO 

30 

6 


=  I 


19200^0  =  3a2 
720^^  =  Zab 
WO  =  V^ 


Ans.  836. 


a  =  80^ 
b  =    30 


19929(9^  =  3a^  +  3ah  + 


2066700  =  3a^ 
14940  :=  3ab 
36  =  P 


a  =  830 
b  =      6 


2081676  =  3a2  +  3ab  + 


In  practice  the  figures  in  italics  in  the  preceding  problem 
are  omitted,  so  that  at  each  successive  step  we  seem  to  be 
dealing  with  tens  and  units.     And  it  often  happens  that 


INDICES;  SURDS;  MOOTS, 


269 


the  first  guess  for  J  is  a  wild  one, —  particularly  if  the  first 
figure  of  the  root  is  small. 


Model  M.      |/56623104. 


56  623  104 

27 


29  623 


27  872 


1  751  104 


1  751  104 


384 


2700  =  3a2 
720  =  3ab 
64  =  b^ 


3484  =  3^2  _|.  3^^  _|_ 


433200  =  3a^ 
4560  =  dab 
16  =  P 


a=  30 
b=    S 


a  ■-  380 
b  =      4 


437776  =  da"  +  Sab  +  ¥ 


Model  N.   i/29.23. 


29.230  000  000 

27 

3.0804 

2  230 

2700  =  3a2 

2  230  000 

270000  =  3^2 
7200  =  Zah 
64  =  b^ 

2  218  112 

277264  =  3^2  +  3«Z>  +  Z*^ 

11  888  000 

28459200  =  Za?' 

11  888  000  000 

2845920000  =  3a2 
3696000  =  3a& 
16  =  &2 

11  398  464  064 

2849616016  =  3^^  +  3«^  +  V 

489  535  996 

a  =  30 

a  =  300 

5  =   8 


a  =  3080 
5  =   0 

a  =  30800 
5  =    4 


270  ALGEBRA. 

EXERCISE    CXLIII. 

Find  the  cube  roots  of: 

1.  51895117.     2.  63044792.     3.  9181846584.  . 

4.  7587307125.    5.  6946005312.   6.  2700. 

7.  54.      8.  100.      9.  2.      10.  20000. 

Cube  Roots  of  Algebraic  Expressions. 

289.  For  cube  roots  of  algebraic  expressions  the  work  is 
just  the  same  in  principle,  but  the  result  can  generally  be 
found  by  inspection — unless  the  polynomial  is  very  long 
indeed — if  one  is  sure  that  the  given  expression  is  a  perfect 
cube.  For  the  first  term  and  the  last  term  of  the  cube 
root  are  respectively  the  cube  roots  of  first  and  last  straight 
products;  and  the  terms  next  to  each  of  those  in  the  root 
can  be  seen  by  dividing  the  next  term  in  the  given  expres- 
sion by  the  ^'  trial  divisor'^  (3^^^). 

Model  0. — The  first  and  last  terms  of 

8«^+12a8_i86i'?+13a6+54a5-51a^-10«3+63a2_54^_|_27 

are  respectively  2a^  and  +3;  starting  from  the  first  term, 
3^2  would  be  3(2^3)2=  12^6;  so  h  would  be  12^^  -^  12^^  =  a^. 
rearranging  the  terms  in  reverse  order,  so  as  to  start  with 
the  last  term,  Sa^  =  3(9)  =  27 ;  then  h  would  be  -  54a  -^  27 
=  —  2a.  The  root  would  be  ^a?  +  a^  _  2a  +  3.  To  verify 
this  result  it  is  easier  to  go  through  the  motions  of  finding 
a  cube  root  than  to  multiply  out  the  third  power. 


INDICES;  SURDS;  BOOTS. 


271 


Model  P. 

S^+12a8-18a^+13a«-f54a« 

2a»  +  a«  -  2a  4-  3 

-51a4+10a3+63a'-54a+27 

12a«  =  3a'        a  ==  2a8 

M^ 

6a5  =  3a5        6  =  a« 

12^8  -f  6a'  +  a« 

a^=:6« 

-24a'4-12a«4-54a5-51a4  +  lO^^ 

12a«  +  12a^  +  3a^  :=  3a2 

+  63a'-54<*  +  27 

-  12a*  -  6a3  ==  3a&       a  =  2a^  +  a' 

4a'=.b'^          b=-  2a 

-24^''-  24a«  +  18«5  -f  12tt4_8^3 

36a«  +  36a5-63<*4  _|_  ig^^  _^  ^3^2 

12a«+12a5-21a4-12a2+12a2=3a2 

-54a  +  27 

18a3  +  9a2-18a=3a6 

9  =  6^ 

86a«  +  36a5  _  63a4  +  18^'  +  63a5 

[a  =  2a»  +  a'  -  2a] 

-  54a  +  27 

[6=3] 

EXERCISE  CXLIV. 

Find  the  cube  roots  of: 

1.  x^  +  96x  -  4:0x^  +  6x^  -  64. 

2.  x^  +  Ux^y  +  QOxY  +  160:^)y  +  2402:y  +  192a:^5 

+  64^«. 

3.  27a:9  _   54^8  _p   53,^7  _  71^6  _^  57^5  _  35^4  ^  22x^ 

-  9x^  +  3x  -  1. 

4.  Sa^  -  12a^  +  lSa'  -  Ua^  +  9a^  -3^+1. 

5.  x^^  -  dx^^y  +  3x^Y  +  '^^Y  -  ^^Y  +  ^^Y  -  ^^Y 

+  ^x'^f'  -  "IxY  -  ^ccY"^  +  ^xf''  -  y'^. 

6.  3285092;6  -  480861:?;^  +  283383^V^2  _  ^06b^h^ 

7.  8x^2  _  i22;ii  +  42a;io  -  Qlx^  +  99^:8  -  117^'^  +  126:?;« 

-  108^5  +  81:^4  -  4:7x^  +  "IXx^  -  ^x  +  1. 

*8.  la^  —  ia¥  +  |«%^  +  Qak^  +  Sb^  —  |a%^  —  I2ab^ 

-  Qah^  +  lah^  +  lah^. 

9.   16^V2  -  72:^«^i/6  +  324a;y4/2  -  1623/3|/6. 

10.  x^  —  dxY^^y^  +  ^xY^^^y  —  y^' 

*  Rearrange. 


272 


ALGEBRA. 


Series  for  Cube  Root. 

290.  As  in  square  root,  if  the  given  expression  is  not  a 
perfect  power,  the  result  will  be  an  endless  series  of  terms. 
Model  a. 


frr 


X, 


1   -  X 

x      x^ 

5x^ 

1 

I-3--9- 

81 

-  .   .   . 

—  X 

3  =  3^2 

a  = 

1 

—  a;  =  3a^ 

b=- 

X 

~  3 

x^ 

27 

-;  =  .^ 

x^        x^ 

~  3  +  27 

3-2.  +  ^^: 

=  3a=' 

«  =  !-! 

x^       2^3 

x^ 

x' 

x^       x^ 
■~  3"  +  9" 

=  3a5 

--1^ 

3  "^  9  " 

81 

729 

x' 
SI' 

=  52 

bx^        x^ 
27  "^  81 

x^ 
^  729 

x^ 
3  -  2^  -  -3 

2a;8 
+  9 

+  - 

EXERCISE   CXLV. 


Fi7id  series  for : 

H  +  X. 


2.     yl  +  2^  +  :z;2. 


3.     VS 


V2>x. 


4.     1^1  -  x^. 


5.     VI  —  X  +  X^, 


Fourth  and   Higher  Roots. 

291 .   By  utilizing  in  the  same  way  the  identities 

(a  +  by  ~a^+  ia^  +  Qa^^  +  4:ab^  +  b^ 
(a+by  =  a^  +  6a^b  +  lOa^^  +  lOd^b^  +  ba¥  +  h^ 
and  so  on. 


INDICES;  SURDS;  MOOTS.  273 

it  would  be  theoretically  possible  to  construct  a  rule, 
analogous  to  those  for  square  and  cube  root,  which  would 
serve  to  find  any  desired  root.  Practically,  such  rules 
would  be  difficult  of  application,  and  of  no  possible  utility 
either  in  numerical  work  or  in  algebra;  because  the  re- 
quired roots  can  be  obtained  more  conveniently  by  other 
means,  as  will  be  explained  in  Chapters  XIV  and  XV. 


CHAPTER  XI. 

LITERAL   EQUATIONS;   GENERALIZATION. 

292.  Referring  to  Model  I  in  Chapter  I,  we  see  that  the 
problem  may  be  stated  as  follows; 

A  man  who  can  row  a  miles  per  hour  in  still  water  finds 
that  it  takes  him  h  hours  to  row  up-stream  to  a  point  from 
which  he  can  return  in  c  hours.  How  fast  does  the 
current  flow  ? 

As  stated  in  Chapter  I  this  problem  had  5  instead  of  a, 
5  instead  of  h,  and  4  instead  of  c]  and  further  on  we  find 
five  other  sets  of  figures,  each  of  which  could  be  substituted 
for  a,  l,  and  c  in  the  statement  of  the  problem  here  given. 
Since  we  have  now  learned  to  perform  all  sorts  of  alge- 
braic operations  upon  letters  as  well  as  upon  figures,  we 
can  take  the  statement  of  this  problem  as  given  just  above, 
construct  our  equation,  and  solve  it,  thus : 
Model  A. 

Let  X  =  the  number  of  miles  per  hour  current  flows ; 
a  -{-  X  =  the  number  of  miles  per  hour  man  rows  down- 
stream ; 
a  —  X  =  the  number  of  miles  per  hour  man  rows  up- 
stream. 


®  h{a  —  x)  =  c{a  +  x) 

(2)  ab  —  bx   =  ac  -\-  ex 

same  as  Q 

@  ah  —  ac    =  bx  -{-  ex 

(z)  -\-  bx  —  ac 

®  ab  —  ac    =  {b  -\-  c)x 

same  as  (3) 

^^           ab  —  ac       aib  — 
^             b  -]r  c            b  + 

c) 
c 

274 

ITEBAL  EQUATIONS;   GENERALIZATION,       275 

293.  An  equation,  like  0,  which  contains  letters  (in- 
stead of  numbers)  to  represent  the  numbers  that  would, 
ordinarily,  be  given  in  the  statement  of  the  problem,  is 
called  a  literal  equation ;  the  answer,  an  algebraic  expres- 
sion, is  in  fact  a  formula,  and  indicates  all  the  arithmetical 
operations  that  must  be  performed  upon  the  given  numbers 
to  get  the  numerical  answer  to  the  problem  for  any  par- 
ticular numerical  case.     Thus  if  the  problem  were  stated : 

A  man  who  can  row  5|  miles  an  hour  in  still  water  finds 
that  it  takes  him  3|^  hours  to  row  up-stream  to  a  point 
from  which  he  can  return  in  2  hours.     How  fast  does  the 
current  flow  ? 
— we  should  substitute  for  a,  I,  and  c  in  the  formula  above, 

51(31  _  2) 
and  get  x  =     ^^  ^    =  3|^  —  2  =  1^  miles  per  hour. 


EXERCISE   CXLVI. 

Similarly,  hy  substituting  in  the  formula,  get  the  values 
of  the  speed  of  the  current  corresponding  to  the  other  sets  of 
values  of  a,  h,  and  c  given  for  this  problem  in  §  26. 

Obtain  a  formula  and  substitute  as  before  for  example  1 
in  Exercise  XL 

294.  The  degree  of  a  term  in  a  literal  equation  is  the 
number  of  unknown  letters  that  appear  in  that  term  as 
factors;  the  other  letters  representing  such  numbers  as 
would  be  numerically  given  in  the  statement  of  the  problem 
to  be  solved  by  this  equation. 

296.  In  literal  equations  of  the  first  degree  with  one 
unknown  letter,  the  known  terms  are  arranged  on  one  side 
of  the  equation,  and  on  the  other  side  the  unknown  letter, 
say  X,  is  then  recognized  and  separated  as  a  distributive 
factor.     The  othee  factor  is  the  coefficient  of  x. 


276  ALGJEBUA. 

Model  B. 

Ix  +  Wx  -\-  a^  =  ZIH  +  Z'^  +  ax. 

a?  —  1)^  =  ax  —  hx  —  3a^x  +  3^^^ 
x{a  -  b  -  3a2  +  3Z^2)  ^  ^s  _  ^3 

a^-  ¥ ci^  j^  ah  J^y^ 

^  ~  {a  -  h){l  -  3a  -  3Z^)  ""  1  -  3a  -  3^> 

EXERCISE  CXLVII. 

X     ,        X             a-\-l)             /    ,  zx     ,     1  ,         z 

1.  — n+^i J.=T~^ — Tz.-    2.  (a+c>)a;H -=:?;-f«— J. 

a;  —  Z>a;~-6;:?;  —  a_2;  —  a        1         1 
c  a      ~^      h       ~~     abc  ac       aV 

4.  ac~  —  ^ -X  —  bx  =  ap  —  36x, 

q  a  ^ 

i-L    I     7x  3(y^  ^Tc)  -hx       1 

6.   {a  ^  x){p  —  x)  =  3{b  —  xf. 
x^  —  Jix  -\-  Jc       x"^  ~  px  -\-  q 


7. 


X  —  h  X  —  p 


(x  -  af    ,    {x  -  by    ,    {x  -  cy      ,^     ,  ,      ^,  a:2 

k  —  x      *      X  +  a         -^ 
X  —  a,x  —  b,x  —  c  xia  -\-  b  -\-  cY 


a-\-b^  b  -{•  c^  c  -^  a      {a-\-b){b  +  c){c  + ay 


LITERAL  EQUATIONS;   OENERALIZATION.        277 

Model  C. — To  pay  a  sum  of  n  dollars,  p  coins,  some  of 
value  a  cents  each  and  the  rest  of  value  l  cents  each,  are 
found  to  be  sufficient.     How  many  coins  of  each  kind  ? 
Let  X  =  the  number  of  a-cent  coins; 
p  —  X  =  the  number  of  Z>-cent  coins. 

(\)  ax  -{-  hp  —  hx  =  lOOn 

(z)ax  —  bx  =  lOOn  —  hp 

(D  x{a  —  b)  =  lOOn  —  bp 

^^           lOOn  —  bp,  »      .  ,,         ,         .      , 

(A)  X  =^ 7-^ (no.  of  coms  worth  a  cts.  apiece) 

^5.  100^^  —  bp      ap  —  bp  —  lOOn  +  bp 

®p  —  x  —p =-^  =  -^ ^ ^ '—^ 

^^  ^  ^  a  —  b  a  —  b 

_ap—  lOOn, 


a  —  b 


-(no.  of  coins  worth  b  cts.  apiece) 


EXERCISE    CXLVni. 

1.  A  father  is  a  times  as  old  as  his  son ;  p  years  ago  he 
was  b  times  as  old.     What  are  their  ages  now  ? 

2.  There  is  a  difference  of  q  cents  a  pound  in  the  price 
of  two  commodities;  a  pounds  of  one  and  b  pounds  of  the 
other  amount  to  h  dollars.  Price  of  each  commodity  per 
pound  ? 

3.  Paid  s  dollars  with  coins  worth  a  cents  and  coins 
worth  b  cents ;  d  more  coins  of  the  first  kind  than  of  the 
second.     Number  of  coins  of  each  kind  ? 

4.  I  have  m  dollars,  in  nickels,  dimes,  and  3-cent  pieces ; 
a  times  as  many  dimes  as  nickels,  b  times  as  many  3-cent 
pieces  as  dimes.     Number  of  each  ? 

6.  A  merchant  has  grain  worth  a  cents  per  peck  and 
other  grain  worth  b  cents  per  peck;  in  what  proportion 
must  he  mix  m  bushels  so  that  the  mixture  may  be  worth 
q  dollars  in  all  ? 


278  ALGEBRA, 

6.  A  man  who  can  row  5  miles  per  hour  in  still  water 
finds  that  he  can  row  down-stream  in  a  hours  to  a  point 
from  which  it  takes  him  h  hours  to  return.  How  fast  does 
the  current  flow  ? 

7.  A  man  who  can  row  a  miles  per  hour  finds  that  it 
takes  him  r  times  as  long  to  go  up-stream  as  to  go  down. 
Find  the  speed  of  the  current. 

8.  Bought  cards  a  for  a  nickel,  and  the  same  number  of 
other  cards  J  for  a  nickel;  then  sold  them  {a-\-l))  for  a 
dime,  and  lost  p  cents.     How  many  cards  were  bought  ? 

9.  Walking,  a  certain  distance  can  be  accomplished  in 
a  hours;  riding,  in  h  hours;  walking  half  the  time  and 
riding  the  other  half,  how  long  does  it  take  for  that  dis- 
tance ? 

10.  The  fore  wheel  of  a  carriage  has  a  circumference  of  a 
feet,  the  hind  wheel  of  l  feet;  when  the  fore  wheel  has 
made  c  turns  more  than  the  hind  wheel,  how  far  has  the 
carriage  gone  ? 

11.  A  can  do  a  certain  job  in  a  days,  B  in  Z>  days;  how 
long  for  both  ? 


Factoring  Literal  Quadratics. 

296.  In  solving  literal  quadratics  the  equation  should 
first  be  arranged  in  the  standard  form 

ax^  ~\-  Ix  -\-  c  =  0 

and  then  factored  by  the  method  of  cross-multiplication.* 

*  In  discussing  literal  equations  it  is  important  not  to  confuse 
the  letters  a,  b,  and  c  which  represent  the  three  coefficients  of  the 
quadratic  with  similar  letters  which  may  occur  in  the  statement  of 
the  equation  itself. 


LITERAL  EQUATIONS;   GENERALIZATION,        279 

Model  D.     Solve  x  -  o?  =  '^0' -i- V' -  -^^,. 

x-\-  a^ 

®  x^-'^o?x-Wx^Za^-\-Za''¥      ©  -Id'x-Wx-'itd'-^Za^^ 
®  a;  -  3^2  =  0  I    from  ® 

(5)  ^  +  (^2  _  ^2)*^  0  )     by  Ax.  A.     (See  Chapter  Y.) 
@x  —  36^2.  ^.  _  ^2  __  ^2^     ^^^^^ 

EXERCISE   CXLIX. 

1.  x^  -  (3^  -  4)i?;  +  2^2  _  5a  +  3  =  0. 

2.  :^2  +  (2a  +  V)x  -]-  a^  ^  ah  -  W  =  0. 

3.  x^  -^  {a  -  h)x  -  (6^2  4-  I3a^>  +  6^>2)  ^  0. 

4.  a;2  4-  7:^;  4-  10  =:  4.ax  +  17a  -  Sa^. 
x^  -  {a?  +  13Z^2) 


6.  ^2  -  60(^2  -  J2)  ^  ^(196  _  11^). 
a^  -\-  a  — 
1  +  X 
a{a^  —  x) 


2        a^  +  a  —  1 

7.  a;  +  a2  ^  ±- 

1  +  ^ 


%,  X  -\-  1  — 


a  -{-  X 


2       ^ 
or 

12a  ~^         9        ~  ~T6 


X  +  a^  I      ,    137a2 

10.  -J-  =  J^  +  -IT 


297.   The  operation  of  factoring  quadratics  becomes  more 

difficult  when  neither  straight  product  is  a  monomial. 

Model  E. 

(9^»2  -  4a2)a;2  +  {4:ah  -  4.a^)x  -  {a^  -  2ah  +  J)")  =  0. 

{Sh  +  2a)x         +  (a  -  h) 

(di  —  2a)x         —  {a  —  h) 

h  —  a  h  —  a  . 

X  = ^       Ans, 


2a  +  W  2a  -  U 


280  ALGEBRA. 


EXERCISE  CL. 


1.  (a2  ~  V^)x'  -  2{a^  +  i^)x  +  ^2  _  52  =  0. 

2.  {a^  +  ^a  +  2)x^  4-  5a;  +  2  =  2a^x  +  3a  -  a^. 

8.  a\2x^  4-  3a;  +  1)  -  Z>''^(a;2  +  4:X  +  4:)  =  ab{x^  +  2x). 
4.  a\x^  -  1)  =  «^(2a;2  _  7a;  -  1)  +  ^>2(3a;2  _|_  5^  _  2). 

^2(2a;  +  3)  __  d\2  +  3x)  __        ah{x^  -  1) 
^'      1  +  2a;         ^   a;  +  2      ~  (a;  +  2)(1  +  2a;)* 

a2(5a;  +  3)       h\3x  +  5)  _         ah{x''  -  1) 
^-      3  +  2a;     ""     2  +  3a;     ~  (3  -r  2a:)(2  +  3a;)" 
7.  x{iox+ll){a'+h'')-3lal{x^-\-l)z=:3ba'+h\(6-2^x'). 

{x^  -  l){a^  +  ah  +  P)  _    Sab  alx 

(a  +  ^)^  ~  a  —  1)~  a  -\-  V 

9.  2a(l  -  10a;2)  +  2  +  ha^x?  =:  (8  -  5a  -  76^^)^, 
«J  +  6  2«(a;2  +1)  x^  -\-  a  -I  -\ 


10. 


a  +  2>  ~  (a  +  l){\  -  a;2)  a;2  -  1 


Solving  Literal  Quadratics  by  Formula. 

298.  Equations  which   are  very  difficult  to   factor   by 
inspection  may  be  solved  by  the  quadratic  formula. 

Model  F. 
2«*-5a3-27+15aa;(a2+3)::z2«a;2(a34_^_|_10)4-{17a2_i8):z; 
Here  a  =  2a^  +  2a^  +  20a 

J  =  _  15^3  +  Yia"  -  45a  -  18 
c  =  _  2a^  +  ha^  +  27 

J2  -  4ac  =  16a8  -  40a^  +  241a«- 390aS+ 1023a* -QOOa^ 
+  197a2  -  540a  +  324 


4/^2  -  4ac  =  4:a^  -  ba^  +  27a2  -  15a  +  18 


LITERAL  EQUATIONS;   GENERALIZATION.       281 


__  —h±V¥—4:ac 
^"  ¥a 

2^3  +  a2  _|_  3^   I  9  3  _  ^ 

^  =  -irV — A  \   .   -.A — 5  or    a;  =  — — ^.     ^7i5. 


EXERCISE  CLI. 

1.  («»  -  U^)x^  +  10«a;  -  (4a3  -  7a  +  3)  =  0. 

2.  aV  +  {d^  +  2a):^;  +  a^  +  6  =  Saa;^  +  (2^2  +  9):z;  +  7a. 

3.  (2a3+  5a2)a;2+  (Sa^-  2)a;  =  2a3-  5a2+  4:  +  ax  -  2axK 

4.  (a^  -  1)^2  +  (4a3  +  4)ic  +  3a^  +  lla^ 

3=  4  +  (a2  -  a)ir2  +  (2a  -  4a2):r. 

5.  (3a2  -  4)  +  (2a2  +  3a  +  2)x  =  {a^  +  a^)x^  +  a^+'^a^x. 

6.  a\x^  4-  4a;  +  3)  -  (9^;  +  5)  i?;  =  x{x  -  4a2)  +  25a  +  6. 

7.  2(2a;-l)(:r+2)4-a2(2iz;+a)2=-2a2^-a(4a;2+2ic+3). 

8.  a\2x^  +  5^  +  3)  -  (9a  -  %)x^  -  143a  +  28 

=  X  +  (67a  -  ba^)x. 

9.  a\%ax^  -  3)  +  (5a2  +  7a-2)iz;  =  2(a4+a2;2)-5-7a3i«;. 
10.  2a3(a  +  x^)  +  %x{ax  ~  1) 

=  11a  +  10  -  a;(4a^  +  ^a^  +  14a2  +  14a). 

299.  With  practice  even  these  may  be  factored  by  in- 
spection. In  Model  F,  a  and  l  may  be  presumed  to  have 
integral  factors  which  can  be  found  by  trial;  the  factors  of 
the  straight  products  having  been  found,  it  should  then  be 
possible  to  arrange  them  so  as  to  give  the  correct  cross 
products.  For  the  expert  student  it  may  safely  be  said 
that  the  most  economical  way  to  factor  a  literal  quadratic 
is  by  inspection. 


282  ALGEBRA. 


Literal  Simultaneous  Equations. 

300.  Model  a 

®  (^x^  +  'b.y  =  c^ 
®  ci,^  +  Ky  =  0^ 

(D  a^a.x  +  aj)^y  =  a^c^  ®  X  ^, 

®  a^a^x  +  afi^y  =  a^c^  ®  X  a, 

®  {(^A -  (^A)y  =  «i^a  -  ^.Oi     ®  -® 


Similarly  we  obtain 


<^A  -  <^A 


It  is  generally  cheaper  to  eliminate  for  each  letter  inde- 
pendently than  to  substitute  a  complicated  literal  expres- 
sion for  one,  to  find  the  other. 


EXERCISE 

CLII. 

1. 

c,x  +  8^y  =  p,. 

6. 

X  +  ay  =  h. 

c,x  +  s,y  =  p,. 

hx  -{-  y  =  a. 

2. 

x-^  a,y  ^  1. 

7. 

ax  -\-  h  =  y. 

a^x^y  ^  1. 

X  -^  y  =  a. 

8. 

y  =  ^:«  +  ^1. 

8. 

^  +  y  =  (^,' 

y   =  ^,^  +   ^a- 

hx  -^  y  =  a^. 

4. 

a        b 

9. 

a,       5, 

x  +  y  =  a. 

-  +  f-  =  l- 
a,      I, 

5. 

X  +  y  =  a. 

10. 

a{l^x^)=:x^'\a  +  \). 

X  -  y  =  h. 

ax  —  by- 

LITERAL  EQUATIONS;    GENERALIZATION,       283 

11.  a^x  +  h.y  +  c,z  =  h,  \ 

a^x  +  ^^y  +  o^z  —  Ic^K  Find  x  only. 

12.    a^  +  Z>2/  +  ;2  =  6'.  13.    a;  +  ^^  =  ^. 

^  +  ^y  +  ^^  =  ^-  y  -\-  hz  =  c, 

bx  -{-  y  -\-  az  =  c.  z  -\-  ex  —  a, 

l^.  X  —  ay  —  h,  i^,  X  -\-  y  -{-  z  =  a,. 

y  —  Iz  —  c,  X  —  y  -{-  z  ^  a^, 

z  —  ex   —  a,  X  —  y  —  z  =z  a^. 

16.   2.  +  ^-.v-^.  +  |.     17.  1  +  ^  +  ^=3. 

y  —  X  +  2z  =  a^  —  ~,  =5. 

«     ,  y  c       a 

^-^,+  y  =  a,--  _^_. 

18.  i^  +  .V  +  ^  =  ^'  19.  i^^  +  2y  =  «  +  45  +  3. 

i<;  — y=  ^j  —  2a^  +  ^3-  y  4"  ^^  ==  ^^  +  ^^  +  2- 

^+^!/+^^==<^3~"^^i"~^2-      z  -\-  '2x  —  2a  +  3c  +  3c 

20.  a;  +  2/  +  ;<;  +  ^^  =  2«  +  ^5- 

X  —  2y  -\-  z  =^  b  —  a  ^^  w  -\-  'dy  —  2a. 
X  -{-  y  —  z  —  w  ~  2a  —  2b  —  2c. 

2x{3y  -j-  x)  _a 


21.  2x  +  3y  =  a; 


9 


^+3^         4 


a+b  X  o      ,    A 

''■~^  =  ^P^'  2^  +  4^  =  «- 

-,  X?  —  ab      y  +  h 

23.  a;  +  a  —  5  =  ;?/; =  "^    '     . 

^        y  —  a  2 

a  +  b        2a  +  b      2y  +  3a 

24.  X  —  3a  =  2y\     5—^7 ^^— -  =  -^—^ . 

-^^     2y+-a^       a  a 

25.  ^  =  n^  +  Z>  +  ^:     -^^ +  ^- =  2. 

^  '       '  2/  y  —  ^ 


284  ALGEBRA, 

a  -\-  ^h       y  —  X  _x  -\-  a 


X  -^  a  —  4:b; 


X  -{-  )lb      X  —  2b 


a        b  X       b 

2%.  X  -{-  y  ^^  ba]     xy  \b  ^=^  ha  —b, 

29,  x?"  -\-  axy  +  ay'^  =  xy ;     x  Ar  y  ^=^  Sa, 

30.  x^  -{-  y^  =^  a^xy ;     x  -{-  y  =  bxy. 

Constants  and  Variables. 

301.  In  the  Chapter  on  Elimination  we  saw  that  in  the 
equation 

2x  —  3y  =  6, 

while  X  and  y  are  restricted,  still  the  value  of  x  may  vary 
from  any  negative  number  to  any  positive  number;  and 
the  value  of  y  changes  with  every  value  of  x. 

Similarly  in  the  equation  ax  —  by  =^  c  the  values  of  x  and 
y  could  vary,  subject  only  to  the  restrictioii  that,  whatever 

the  value  of  x,  y  must  be  given  by  the  formula  y  =  — 7—  ; 

and  a,  b,  and  c,  like  the  2,  3,  and  6  in  the  numerical 
equation,  remain  unchanged  throughout. 

302.  In  any  problem  the  numbers  whose  values  are 
given  are  called  constants  of  the  problem;  and  those  whose 
values  are  required  are  called  variables. 

Thus  in  Model  A  of  this  chapter  the  numbers  «,  b,  and  c 
are  the  constants;  the  (unknown)  speed  of  the  current  is 

the  YARIABLE. 

As  the  speed  of  the  current  is  supposed  to  have  a  certain 
fixed  value,  which  we  are  trying  to  find  when  we  start  to 
solve  the  problem,  it  seems  strange  to  call  that  speed  a 
VARIABLE  ;    but   wc   may   state   for   this   problem    three 


LITERAL  EQUATIONS;    GENERALIZATION,        285 

equations  of  condition,  in  any  one  of  which,  taken 
separately,  x^  or  y,  or  z,  is  a  variable,  subject  to  a  restriction. 
If  we  let  y  =  the  speed  of  the  boat  down-stream,  and 
z  =z  the  speed  of  the  boat  up-stream ;  then  y  =  a  -{-  x  ; 
z  =  a  —  x;  bz  =  cy.  So  far  as  the  first  of  these  equations 
is  concerned,  x  may  have  any  value,  so  long  as  y  is  greater 
than  it  by  a.  But  no  algebraic  restatement  can  make  a,  or 
b,  or  c  appear  to  be  vaeiable  ;  they  are  the  constants  of 
the  problem. 

Discussion  of  Problems. 

303.  A  problem  is  said  to  be  generalized  when  one  or 
more  letters  are  used  instead  of  numerical  constants. 

304.  The  investigation  of  a  generalized  answer,  with  a 
view  to  determining  the  effect  upon  it  produced  by  different 
values  or  relations  imposed  upon  the  constants,  is  called 
Discussion  of  the  problem. 

LIMITING  VALUES. 

305.  In  discussing  a  generalized  problem  we  often  meet 
values  which  by  the  ordinary  rules  of  algebra  have  no  mean- 
ing whatever.  The  two  most  important  of  such  results 
are,  first,  that  obtained  when  the  denominator  of  a  formula 
becomes  0  for  a  particular  set  of  values  for  the  constants; 
and  second,  that  obtained  when  the  numerator  and  de- 
nominator both  become  0.  It  is  usual  to  consider  the 
following  three  cases  together : 

(i)   Where  the  numerator  is  0; 
(ii)   Where  the  denominator  is  0; 
(hi)  Where  both  numerator  and  denominator  are  0. 

These  are  called  limiting  values  of  the  formula  under 
discussion. 


286  ALGEBRA. 

Model  H.-^If  we  set 

_  x^  —  Sx  -{-  15 
y  ~  x^  -  Ix  +  12 
we  obtain  the  following  limiting  values : 

25  -  40  +  15  _  0 


(i)  X  =  b]  y  = 

(ii)  X  =  ^]  y  :=^ 

(ill)  a;  =  3 ;  2/  = 


25-35  +  12       2 
16  -  32  +  15  _  _  1 
16  -  28  +  12  ~        0 
9-24  +  15  _  0 
9-21  +  12  ""  0 


In  considering  (i)  we  may  conclude  at  once  that  since  if 
0  is  divided  into  2  parts,  the  result  will  be  zero;  and  since 
if  2  is  divided  into  zero,*  the  quotient  and  the  remainder 

will  both  be  zero ;  then  the  value  of  -  must  be  0. 

When  we  come  to  (ii)  we  have  no  such  easy  task,  since 
to  divide  1  by  0  is  meaningless;  and  still  more  meaningless, 

in  (III),  IS  -. 
We  have  to  investigate  then  the  three  expressions  (i)  — ; 

(")o;  (™)o- 

Let   us   consider   at  first   the  three   expressions  (i)  — ; 

(ii)  — ;    (ill)  — ;  where  x  and  y  are  variables,  and  a  and  h 
are  positive  constants. 

X 

(i)  So  long  as  x  is  positive,  —  will  be  positive;  as  x 
decreases,  —  decreases  also;    when  x   passes  over  from  a 

*  As,  for  example,  wlien  one  is  asked  how  many  times  lie  can  fill  a 
two  quart  measure  from  an  empty  bin. 


LITERAL  EQUATIONS;   GENERALIZATION.       287 


X 

small  +  value  to  a  small  —  value,  —  does  likewise.     At  the 
instant  when  x,  in  its  progress  from  +  values  to  —  values, 

X 

assumes  the  value  x  =  0,  —  2X  the  same  instant  assumes 

a 

the  value  0. 

We  may  conclude,  then,  that 

a 

and  also  we  may  say  that 

0  X 

(i)  —  is  the  limit  which  —  approaches  as  the  numerical 
^  ^  a  a     ^ 

value  of  X  is  indefinitely  diminished. 

X 

If  a  had  been  a  negative  constant,  then  —  and  x  would 
have  opposite  signs  just  so  long  as  x  was  -\- ;   but  at  the 

X 

instant  when  x  passed  from  -f  to  — ,  —  would  pass  from 

—  to  -f ,  and,  as  before,  we  should  have  —  =  0. 

(11)  If  now  we  suppose  y  to  begin  with  a  considerable 
+  value  and  then  to  diminish,  —  would  not  diminish,  but 

y 

increase,  on  the  principle  that,  the  dividend  remaining  the 
same,   a   smaller   divisor   gives   a   larger    quotient.     If  y 

becomes  — ,  —  also  becomes  — . 

y 

As  y  passes  through  zero,  from  the  region  of  +  numbers 

to  the  region  of  —  numbers,  -  passes  also  from  the  region 

of  +  numbers  to  the  region  of  —  numbers;  but  it  does 
KOT  pass  through  0.     We  conclude  then  that 

(11)  ^,  like  0,  is  a  boundary  between  -}-  and  —  numbers. 


288  ALGEBRA. 

Again,  by  causing  y  to  assume  a  very  small  value,  posi- 
tive or  negative,  and  to  become  successively  smaller  and 

smaller,  we  cause  -  to  assume  a  very  lakge  value  positive 

or  negative,  and  to  become  successively  larger  and  larger. 

By  choosing  y  suitably  we  can  cause  —  to  assume  a  value 

greater  th^n  any  assignable  value,  however  large;  and  still, 

if  y  is  then  taken  weaker  to  0  in  value,  —  becomes  even 

^  ^    y 

larger.     Hence  we  also  conclude: 

(ii)  ?r  is  a  limit  towards  which  large  positive  or  large 

negative  numbers  may  approach  if  we  suppose  them  to 
increase  indefinitely. 

The  symbol  for  this  limit  is  oo  and  its  name  is  infinity. 

(n)^-co. 


(hi)  If  we  suppose  the  expression  —  equal  to  any  con- 
stant h,  we  could  still  allow  x  to  vary,  and  to  assume  any 
value  whatever,  positive  or  negative;  y  would  always  be 
determined,  for  any  particular  value  of  x,  by  the  formula 

X 

y  =  --.     AYe  may  cause  both  x  and  y  to  assume  values  less 
than    any   assignable   quantity,    however    small,    without 

X 

destroying  the  equation  —  =  k.     In  other  words,  we  may 

X  0       . 

cause   —  to  approach  as  near  as  we  please  to  —   without 

X 

destroying  the  equation  —  =  h.     These  considerations  will 
all   hold    good  whatever  the  value  of  k  ;   hence  we  may 


LITEBAL  EQUATIONS;   GENEBALIZATION,        289 

conclude  that  the  symbol  —  does  not  of  itself  determine 
any  value. 

(hi)  jt  is  indeterminate. 

This  conclusion  does  not  mean  that  —  =  aky  num- 
ber ;  only  that  when  substitution  in  a  formula  leads  to  this 
result,  some  special  reduction  must  be  resorted  to  for 
determining  its  actual  value,  if  the  value  can  be  determined 
at  all. 

Model  H. — In  the  expression 


^  ~  x^  -lx+  12 

.-     _  ^       _  9-24  +  15  _  0 
'^  ^  ~  "^^  ^  ~  9  -  21  +  12  ~  0  ' 

_x?  —  ^x-^1^  __{x  —  Z){x  -  5)  _a;  -  5  _  , 
^  ""  x^  -7^  +  12  ~{x-  3)(^  -  4)  '^^^^~  ' 


Problems  for  Discussion. 

306.  Model  I. — A  cistern  can  be  filled  by  one  pipe  in  12 
minutes,  and  emptied  by  another  in  36  minutes.  How 
long  will  it  take  to  fill  it  with  both  pipes  open  ? 

Generalized  statement : 

A  cistern  can  be  filled  by  one  pipe  in  a  minutes,  and 
emptied  by  another  in  b  minutes.  How  long  will  it  take 
to  fill  it  with  both  pipes  open  ? 

X  =  number  of  minutes  to  fill,  both  pipes  open. 

—  =  fraction  of  cistern  filled  by  first  pipe  in  one  minute. 


290  ALGEBUA, 

—  =  fraction  of  cistern  emptied  by  second  pipe  in  one 
minute. 

^1        1        1 

^•^  a        0        X 

@  bx  —  ax  =  ab  ©  X  abx 

®  x  =  ^^  ®^{b-a) 

Now  if  b  is  greater  than  a,  this  answer  is  positive,  and  its 
meaning  is  natural  and  obvious.  Thus  in  the  statement  of 
the  problem  first  given,  «  =  12,  J  =  36;  then 

X  =  :^ ^  ^  —  18  minutes  to  fill. 

36  —  12 

If  b  is  less  than  a,  the  answer  is  negative;  on  reflection, 
we  see  that  if  the  outlet  emptied  in  less  time  than  the  supply- 
pipe  filled  (the  emptying  would  then  be  more  rapid  than  the 
filling),  it  would  be  vain  to  expect  the  cistern  to  fill  with 
both  open.     Assuming  a  =  24,  b  =  W,  we  obtain 

24  X  16 


16  -  24 


=  -  48. 


We  miglit  say  that  wliile  in  the  first  case  the  instant  of  being  full 
is  18  minutes  after  the  instant  of  being  empty,  in  the  second  case 
the  instant  of  being  full  is  48  minutes  before  the  instant  of  being 
empty. 

If  b  were  negative,  the  numerator,  ba^  product  of  unlike 
factors,  would  be  — ;  the  denominator,  b  —  a,  sum  of  two 
negative  terms,  would  also  be  — ;  so  x  would  be  plus. 

Now  for  J  to  be  a  negative  number  is  the  same  as  to  say  fill  in- 
stead of  empty;  and  with  both  pipes  filling,  the  answer  would  evi- 
dently be  -j-  and  reasonable.     Thus,  if  a  =  28,  6  =  —  21, 

-  ^8  X  (-  21)  _       28_X  21  _ 
^~    -  21  -  28   ~  "*■       49"     ~ 


LITERAL  EQXIATIONB;   OBNEBALTZATION.       291 

If  a  is  negative,  the  numerator  ah  will  be  negative  and 
the  denominator  l  —  a  will  be  positive;  so  x  will  be  — . 

But  for  a  to  be  —  is  to  have  the  supply-pipe  also  an  outlet ;  and 
in  that  case  (as  before  when  J  <  a)  a;  is  —  and  the  cistern  is  being 
emptied  instead  of  filled. 

If  a  =  Z>,  X  assumes  a  limiting  value;  t  —  a  —  0  and 

ah  a^ 

h  —  a       0 

In  this  case  one  pipe  empties  as  fast  as  the  other  pipe  fills,  and  the 
cistern  will  never  be  any  fuller  than  it  is  now. 


EXERCISE  CLIII. 

In  a  similar  way  generalize  and  discuss  the  following 
prohlems  : 

1.  A  boat  travelling  10  miles  per  hour  passed  south  by- 
Highland  light  at  noon;  a  steamer  pursuing  at  13  miles 
per  hour  passed  on  the  same  track  3  hours  later.  When 
will  the  steamer  overtake  the  boat  ? 

2.  Of  two  pipes  running  into  a  cistern  each  could  fill  it 
in  21  minutes;  the  waste-pipe  could  empty  it  in  35  minutes. 
How  long  would  it  take  to  fill  the  cistern  with  all  pipes 
open  ? 

3.  A  person  who  can  row  5  miles  an  hour  in  still  water 
rows  21  miles  down-stream  and  back  in  10  hours,  the 
stream  flowing  uniformly  all  the  time.  How  many  miles 
an  hour  does  the  stream  flow  ? 

4.  How  much  rye  at  50  cents  a  bushel  must  be  mixed 
with  50  bushels  of  wheat  at  80  cents  a  bushel  to  make  a 
mixture  worth  70  cents  a  bushel  ? 

5.  A  bicyclist  starts  a  ride,  going  22  feet  per  second; 
half  a  minute  later  a  second  rider  starts  60  yards  behind 
the  first  man's   mark,  riding   25  feet  per  second.     How 


292  ALGEBRA. 

far  from  the  first  man^s  mark  will  the  riders  be  side  by 
side? 

Discuss  the  solutions  of  the  examples  in  Exercise 
GXL  VI I L 

DISCUSSION  OF   EQUATIONS. 

307.  In  the  pair  of  linear  equations 

(\)  ax  -^  ly  =  c 

@  px  -{-  qy  =^  r 
we  obtain  by  elimination 

cq  —  hr  ar  —  pc 

aq  —  hp^  «^  —  Ip  ' 

Of  these  two  answers,  x  —  0  ii  cq  —  hr  =  0  and 
aq  —  hp  4^  0  * ;  ^  =  0  if  ar  —  ^c  =  0  and  aq  —  hp  ^  0. 
If   aq  —  dp  —  0    and    cq  —  hr  ^   0,     x  =  <X)  \    and   if 

aq  —  hp  —  ^  and  cq  —  hr  =  0^    x  =  ~,  that  is,  its  value 

is  undetermined  by  these  equations. 

308.  In  considering  the  special  cases  just  described, 
what  we  have  to  decide  is  whether 

{1)     aq  —  hp  =  0 

(2)  cq  -  hr  =  0 

(3)  ar  —  pc  =  0 

From  these  three  conditions  we  get 
aq=^hp\ 
a  _l    y  from  (1) 

p  ~'q    ) 

c        h 
and  similarly         -  =  —      from  (2) 

and        ~  =:  -      from  (3) 
p      r  ^ 

*  The  sign  ^  means  "  is  not  equal  to." 


LITERAL  EQUATIONS  ;   GENERALIZATION.       293 

It  is  evident  that  if  any  two  of  these  conditions  are 
satisfied,  the  third  must  be. 

309.  Let  us  now  suppose  that  only  one  of  these  con- 
ditions is  satisfied.     Suppose 

(1)     aq  —  dp  =  0;     i.e.     -  = — 

Two  equations  that  satisfy  this  condition  are 

Model  J.  ®  3a;  +    ly  =  5 

(2)6x  +  Uy  =  6 


Here 


a       3       1 

^       14      2 

70  -  42       28 
'^       42-42       0  ' 

18  -  30 

y-     0     = 

12 
0 

that  is,  X  =z  CO  ;     y  =  CO 

If  we  drew  a  diagram  representing  the  infinite  lists  of  answers  to 
these  equations,  as  in  Chapter  VI,  we  should  find  them  to  be  a  pair 
of  PARALLEL  Straight  lines,  which  do  not  intersect,  or,  in  other 
words,  which  intersect  at  an  infinite  distance. 

These  two  equations  are,  in  fact,  i:NrcONSiSTEKT;  for 
from  the  first  we  have  (T)  3x  -\-  7y  =  5;  and  from  the  sec- 
ond 3^;  -f-  7^  =  3  @  ~  2 ;  and  dx  -\-  ly  cannot  at  the  same 
time  be  equal  to  3  and  also  to  5. 

310.  If  two  (and  therefore  all)  of  the  three  conditions 
are  satisfied,  we  should  have  a  pair  of  equations  that  would 

KOT  be  iKDEPEi^DEKT.     For  in  that  case  —  =-=—;  that 

p       q       r 

is,  one  equation  would  be  the  same  as  the  other  except  for 

being  multiplied  through  by  some  constant. 

Model  K.  0    2a;  +    3«/  =:  5 

©  lOx  +  15?/  r=  25 


294  ALOBBBA, 


_  75  -  75  _  50  -  50  _      _0 

^  -  30  _  30'     ^  ~  30  -  30'     ^  -y  -  0' 

The  diagram  for  each  of  these  equations  would  be  the  same 
straight  line. 

311.  The  discussion  of  a  generalized  set  of  equations 
with  more  than  two  unknown  quantities  is  very  much  more 
complicated,  and  cannot  be  treated  here. 


Discussion  of  the  Quadratic. 
312.  The  generalized  quadratic  equation 

ax^  -{-ix  -\-  c  =  0 

whose  solution  is  familiar 


presents  important  features  for  discussion. 
Two  values  of  x  are  given  by  the  formula : 


^^ 2a '     ^= 2a ^ 

If  P  —  4,ac  is  +,  <^  and  /?  are  both  real,*  and  different. 
Thus  in  bx^  +  lx  -1  —  0 

^  = 10 • 

If  h^  —  4iac  =  0,      and  y^  are  real,  and  alike. 
Thus  in  %x^  +  30a;  +  25  =  0 

-  15  ±  >/0  15 


18  18* 

I.e.,  not  imaginary. 


LITERAL  EQUATIONS;   GENERALIZATION.       295 

If  W'  —  4tac  is  — ,  Vh^  —  4:ac  will  be  imaginary,  and  the 
values  of  a  and  /3  will  be  conjugate  imaginaries.* 
Thus  in  2a;2  +  3^  +  5  =  0 


-  3  ±  |/-  31 
X  = . 


EXERCISE  CLIV. 


State  the  character  of  the  roots  {whether  real  or  imagi- 
nary, whether  equal  or  different)  in  the  following  equations, 
without  solving  them  ;  and  in  each  case  give  reasons  : 

1.  100a;2  -^Sx+  .001  =  0.       e.f  ^^  +  ^p'^  =  ^P^- 

2.  x^  +  2x+S=zO.  t,\  p=:X  -  x\ 

3.  x^  +  2x  —  3  =  0.  8.t  x^  +p  =  0. 


4.t  px^  +  qx  -  r  =  0.  9.  1  -  VSx^  _  2^  +  3  =  0. 

9        ^  16 -X 

28  =  ^'  '''-^+T 


5.  7^2  ^  3^  ^  ^  ^  0.  10.   -^^  ,    f  -  lOo;  =  15. 


Zero  Coefficients  in  the  QuadratiCn 

313.  If  c  becomes  0,  while  a  and  b  do  not, 


-b+Vb^- 

■  o_ 

0  _ 
2a^ 

h 
a 

0 

2a 

~0 

-b-Vb'- 

2a 

— 

which  are  obviously  the  answers  to  ax^  -\- hx  ^  0. 

*  See  §274. 

\  In  this  example  suppose  p,  q,  and  r  positive  integers. 


296  ALGEBRA. 

314.  If  ^  becomes  0,  while  a  and  c  do  not, 


a  =  — '—- and     p  = 

2a  2a 

which  reduce  to 


Such  would  obviously  be  the  answers  to 

ax^  -\-  c  —  0. 

315.  If  Z>  and  c  both  become  0, 

__   0 
^2a 

-h+  Vb^'  -4.ac      0  +  VO 

=  0 

2a                        2a 

^       -  b  ~  i^b^  -  4ac        0       ^ 
^= 2a =  2-«^'^ 

which  are  obviously  the  answers  to  ax^  =  0. 

316.   It  a  becomes  0,  the  equation  reduces  at  once  to  an 

equation  of   the  first  degree,  I)x-\-  c  =  0^  and  x  —  —  —, 

But  it  is  interesting  to  note  what  becomes  of  the  two 
answers,  a  and  /?,  as  the  coefficient  a  assumes  values  that 
are  very  small  as  compared  with  b  and  c.  When  a  becomes 
YERY  small  indeed  as  compared  with  b  and  c,  the  quad- 
ratic becomes  very  kearly  the  same  as  the  linear  equa- 
tion bx  -\-  c  =^  0^  but  still  has  its  two  answers,  a  and  ^. 
Now  in  ax^  -\~  bx  -\-  c  =  0  ii  a  =  0, 

-  b  +  Vb^      0 
a=  ^— ^- 

-  b  -  V^      --2b 

P  =  7^ =  -7r-  =  «^ 


LITERAL  EQUATIONS;    GENERALIZATION.       297 

The  value  of  a  cannot  be  determined  from  the  formula, 
in  this  case;  but  if  we  rationalize  the  numerator  of  the 
formula  by  multiplying  it  by  its  conjugate  surd,*  we  get 

^l\    \/b''-4cac    i-b-  V¥-4:ac)  _       h^-jh'^-^ac) 


la  '  [-b^  Vb^-4:ac)     '^a{-b-  Vb^-4.ac) 

4:ac  2c 


~2a{-b-  Vb^-iac)  ~~  —  b  —  \/b^  -  ^ac 
If  now  in  this  formula  we  let  a  —  0,  we  get 

2c  c 


a 


-b-b-      b' 


The  meaning  of  these  conclusions  is  this  :  in  any  quadratic  if  the 
coefficient  of  x^  becomes  very  small,  as  compared  with  the  other  two 
coefficients,  one  answer  becomes  very  large,  and  the  other  answer 
becomes  nearly  equal  to  the  value  x  would  have  if  the  first  term  were 
removed  altogether. 

Model  L. — In  the  equation 

\in  z=zl,  we  have  n'  =  —  .502  and  ^  —  —  200  (nearly);  if 
7^  =  2,  a:  =  .5  (very  nearly)  and  §  —  —  20000  (nearly); 
and  the  larger  we  take  n  the  more  nearly  a  becomes  =  .5 
and  the  larger  ft  becomes. 

317.  \i  a  and  b  both  become  0,  while  c  does  not, 


b^  \fV  —  4:ac      0 


2a  0' 

but  by  the  other  formula  a  = ,  =  I^_?  =  oo 

b  +  Vb^'-^ac         0     - 

*  See  §  274. 


298  ALGEBRA, 

and  by  a  second  formula  for  [3,  similarly  obtained, 

—  %G  —2c 

/?  =  -, =^=^  —  — - —  =  00  . 

b-Vb''-4.ac         0 

In  this  case,  if  a  and  b  were  really  zero,  we  sliould  Lave  c  (a  con- 
stant) equal  to  zero  ;  that  would  be,  of  course,  a  meaningless  state- 
ment. Thus-  O.x'^  -\-  0.x  -{-  5  =  0,  being  the  same  as  5  =  0,  could 
not  be  true.  But  if  a  and  b  both  became  veky  small  as  compared 
WITH  c,  it  would  take  larger  and  larger  values  of  x  to  satisfy  the 
equation. 

318.   If  a  and  c  both  become  0,  while  b  does  not, 


2a  0-0 

2o  Q 

by  the  other  formula,   a  = -— =:^  =  --  =  0 

b+Vb^'-4:ac      ^b- 


^.'^     o       -b-Vb^-4:ac      -2b 
while  p  = , =  —7—-  =  00 . 

If  a  and  c  both  become  very  small  as  compared  with  &,  then  a 
will  become  very  small  and  /5  will  become  very  large. 


EXERCISE  CLV. 

Determine  the  limiting  values  of  x  as  n  increases  in- 
definitely : 

2^  1 

1.  10"^2  4-  -—  +  10"  =  0.  2.    ^''x'+  2{3)-x+  -  =0, 
J_U  o 

3    3n^2  _  2»  +  »a;  +  ^„  =  0.  4.  10 V  +^^  +  ±=0. 


CHAPTER  XII. 

PROPORTION   AND   VARIATION. 

319.  An  equation  between  two  ratios  is  called  a  propor- 
tion ;  the  theory  of  such  equations  can  be  investigated  by 
means  of  a  generalized  type,  and  the  laws  so  obtained  ex- 
pressed as  theorems. 

The  general  type  of  a  proportion  may  be  represented  by 

Of  these  four  quantities  x,  y,  a,  h,  x  and  h  are  called 
the  extremes,  y  and  a  the  means;  x  and  a  are  the  anteced- 
ents, y  and  l  the  consequents;  x  and  y  form  the  first  ratio, 
a  and  1)  the  second  ratio. 

The  Three  Simplest  Theorems. 

320.  If  we  assume  0  to  be  true,  equations  @,  (3),  and 
®  are  at  once  obtained,  and  the  theorems  they  express  are 
consequently  true  of  any  proportion. 

I.  The  product  of  the  means  is 
(^hx  —  ay    Q)  X  hy  equal  to  the  product  of  the 

extremes. 


aha 


II.   The  ratio  of  the  antecedents 
is  equal  to  tl 
consequents. 


XV  u 

®-=:  -       (T)  X  —  is  equal  to  the  ratio  of  the 


399 


300  ALGEBRA, 

^  y       h        ,        ^      III.  The  reciprocals  of  the  ratios 
^^  X       a  ^^  are  equal. 


321.  Of  these  three  theorems  the  first  enables  us  to  find 
any  term  of  the  proportion  when  the  other  three  are  given, 
— a  method  known  to  our  grandfathers  as  the  ^^Rule  of 
Three. '^  The  second  theorem  is  known  as  the  ^^Law  of 
Alternation,"  and  the  third  as  the  ^^  Law  of  Inversion/^ 
These  three  are  the  most  elementary  transformations  of  a 
proportion. 

EXERCISE  CLVI. 

2      7 

1.  Find  X  in  the  proportion  —  =  — . 

7 

2.  Find  x  in  the  proportion  {x  —  10)  :  2  = 


3.  Find  x  in  the  proportion 


X  -\-^ 

2^;  +  3  _  a;  +  2 
6  ^dx  —  1' 


4.  By  what  successive  transformations  may  we   obtain 
from  X  :  y  ^=^  a  :  h  the  following  forms  ? 

X  \  a  =  y  :  t 
h  :  y  =  a  :  X 
y  '.h  •=^  X  \  a 

5.  Prove  that  ii  x  :  y  ^  a  :  d,  then  x  :  my  =.  a  :  mb. 


The  Other  Theorems  of  Proportion. 

322.   Keturning  to  the  generalized  proportion 
^  X       a 


PROPORTION  AND    VARIATION.  301 

we  may  represent  the  value  of  each  of  these  two  equal 
ratios  by  some  letter,  say  r. 

^  y       i 
Hence  x  —  ry  and  a  =  rb. 

Then     ®''-+^.^'i±y^':+l=.'Ltl     . 

y  y  1  ^ 

X  -\-  y      r  -{-1 a  -\-  h 


and  (6) 

IV.  In  any  proportion  equal  ratios  are  obtained  by 
dividing  the  sum  of  the  terms  of  each  ratio  by  its  ante- 
cedent or  by  its  consequent.  This  is  called  the  Law  of 
Composition. 

323.  Again  (7) =  — ^  =  -^i —  =  —r- 

y  y  1  ^ 

^^x  —  y      r  —  1      a  —  h 
and  (D ^  =  — ^-  = 


V.  In  any  proportion  equal  ratios  are  obtained  by 
dividing  the  difference  of  the  terms  of  each  ratio  by  its 
antecedent  or  by  its  consequent.  This  is  called  the  Law 
of  Division. 


324.  Again  ®  ?i±^  -  ':^  +  ^  _  M^l  _  a±. 


X  —  y      ry  —  y      r  —  1      a  ~  h 

that  is         — —^  =  — ^■ 
X  —  y      a  —  h 

VI.  In   any   proportion   equal  ratios  are  obtained  by 

dividing  the  sum  of  the  terms  of  each  ratio  by  their  differ- 

-    ence.    This  is  called  the  Law  of  Composition  and  Division. 


302  ALGEBRA. 

325.  Of  the  four  terms  of  any  proportion,  two  are  said 
to  be  homologous  if  they  are  both  antecedents,  both  conse- 
quents, or  both  terms  of  the  same  ratio. 

326.  Homologous  terms,  if  not  terms  of  the  same  ratio, 
may  always  be  made  such  by  Alternation;  and  since  the 
value  of  a  ratio  is  not  altered  by  multiplying  antecedent 
and  consequent  *  by  the  same  number,  we  have  this  trans- 
formation : 

yil.  In  any  proportion,  if  any  pair  of  homologous  terms 
be  multiplied  each  by  the  same  number,  the  resulting 
proportion  is  true. 


327.  Keturning  to  our  first  proportion, — 

VIII.  In  any  proportion,  if  like  powers  be  taken  of  all 
four  terms,  the  four  results  will  be  in  proportion. 


328.  If  we  have  two  proportions 

x^'.y^  —  a^ :  h^ ;  and  x^\y^  =  a^:h^\  then 

IX.  The  products  of  corresponding  terms  will  be   in 
proportion ;  that  is,  multiplying  the  two  equations, 

^A'^i^.  =  «A:^^^.• 
329.  As  in  VI  we  may  prove  that  if 

X      a      p      h       d  _ 
y~b~q~k~s~''' 

then  x^_p^  +  n  +  cl+._^  ^  x_    ^^^^  .^^ 

*  Numerator  and  denominator. 


PROPORTION  AND    VARIATION.  303 

X.  In  any  series  of  equal  ratios,  the  sum  of  all  the  ante- 
cedents divided  by  the  sum  of  all  the  consequents  is  equal 
to  the  ratio  of  any  antecedent  to  its  consequent. 

The  equation  —  =  —  is  sometimes  read  '*  As  aj  is  to  y,  so  is  a  to  5  "  ; 
^  y       b 

or,  '*x  is  to  y,  as  a  to  6'*  ;  and  tLis  sentence  is  the  expression  of 
tlie  statement  that  the  four  quantities,  x,  y,  a,  and  h,  are  in  propor- 
tion; or  that  X  and  y  are  proportional  to  a  and  b.  Instead  of  the  equal 
sign,  the  sign  :  :  was  generally  used;  and  the  proportion  was  some- 
times called  an  "  analogy."  Dropping  this  cumbersome  phraseology, 
much  of  the  difficulty  of  the  subject  of  proportion  goes  with  it,  and 
the  different  theorems  are  seen  to  be  so  many  short  cuts  in  the 
handling  of  a  very  simple  type  of  fractional  equations. 

A  very  complicated  and  difficult  definition,  known  as  Euclid's  defi- 
nition of  proportion,  is  gradually  passing  out  of  use. 

EXERCISE  CLVII. 

Assuming  that  x  :y  =  a  -.ly  establish  the  foJloiving  equa- 
tions : 

1.  inx  :  mxy  z=  a  :  hx,  2.  xy  :  ah  =  y^  :  h^, 

3.  Icmx  :  myz  —  ok  :  hz,         4.  px^-{-qy'^:y^—pd^-\-ql)^  :  V^, 

6.  mx  -\-  y  :  ma  -\-  h  =^  px  -^  y  :  pa  -\-  1). 

6.  px  -{-  qy  :  px  —  qy  =  pa  -\-  qb  \  pa  —  qh, 

7.  x^  +  2/  :  ^2  _^  2h^  =  {x  -\-  yY  :  {a  +  b)\ 

8.  2a;2  +  3^2  .  ^^z  _|_  3^2  ^  ^y  .  ^2>. 

9.  "^x^  -  3^2  .  2^2  -  W  =  2x^  -  ly^  :  2a^  -  W, 

10.  2:?;2  _  3^.y  _  2^2 .  2^2  _  3^j  _  2^2  =2^^^+qy^'^pct^+qV^' 

Assuming  that  a  \x  ^=^  b  :  y  =  c  :  z,  prove : 

11.  a  -\-  3b  +  be  :  a  —  2c  =  X  +  3y  -{-  6z  :  X  —  2z. 

12.  px  -\-  qy  -{-  z  :  px  +  qz  =  pa  -{-  qb  +  c  :  pa  -j-  qc, 

13.  x^-\-y^—xyz:xy'^-\-yh-\-z^x=a^-\-b^—abc:  aW-\-b^c-\-c^a, 
c?  -^  W  —  be       ac  —  ba 


14. 


x^  +  y^  —  yz      xz  —  xy' 


304  ALOBBRA. 


X  -\-  2y  -\-  z  _  Vx^  +  y'^  -\-~  z^  —  3xy 

Use  the  theorems  to  reduce  the  folloiving  equations  : 

16.  Apply  to  the  proportion 

2a+  bh  +  c  +  U  _  2a  +  bh  -  c  -  M 
'^a  —  bl)  +  c  —  M~  "^a  -  bt)  —  c  -\-  M 

the  theorems  VI,  II,  YI  successively. 

17.  If  2;  -h  2 :  7  =  y  +  3 :  9,  find  the  ratio  oix-bioy  -0^. 

18.  Apply  the  law  of  Composition  and   Division  to  the 

^X^+2X+1  '^X^   —   X+1  r^, 

equation  |^rz:^^-^  =  2^M^^-^-     ^^^^  ^^^^^ 
for  X. 

19.  Given  x^  +  1  :  2x'  =:  97  :  72;  find  x. 

20.  Solve  2a;2+10a;+53 :  18a:+  45  =  101 :  99.     (Apply  VI.) 

330.  A  proportion  may  be  formed  out  of  three  quanti- 
ties if  the  first  divided  by  the  second  gives  the  same  ratio 

.  as  the  second  divided  by  the  third.  In  that  case  the  sec- 
ond quantity  is  called  a  mean  proportional  (or  geometric 
mean)  between  the  other  two;  and  the  third  quantity  is 
called  a  third  proportional  to  the  first  two. 

In  any  proportion  the  second  consequent  is  called  a 
fourth  proportional  to  the  other  three  quantities. 

For  a  mean  proportional  the  order  of  the  extremes  is 
immaterial;  for  a  third  or  fourth  proportional  the  other 
terms  must  be  given  AS  they  are  arrakged  ijs"  the 

PROPORTION". 

331.  When  terms  in  a  series  are  so  chosen  that  the  ratio 
of  any  consecutive  pair  is  the  same  as  the  ratio  of  any 


PBOPORTION  AND    VARIATION.  805 

other,  the  quantities  are  said  to  be  in  continued  proportion 
(or  in  geometric  progression). 

Thns  16;8;4;2;l;|;i;  etc.,  are  in  continued  propor- 
tion ;  so  are  a,  J,  c,  d,  e,  /,  .  .  .  if 

a  \  h  =^  h  :  c  =^  c  \  d  ^^  d  '.  e  ^=^  e  :  f  =^  ,  .  . 

Of  a  series  of  terms  in  continued  proportion  any  term 
is  a  mean  proportional  of  the  two  terms  adjacent  to  it. 
Thus  in  the  two  examples  cited,  4  is  a  mean  proportional 
between  2  and  8;  c  is  a  mean  proportional  between  b 
and  d.     (See  further  in  the  chapter  on  progressions.) 

EXERCISE  CLVIII. 

1.  Find  a  fourth  proportional  to  2,  3,  and  4. 

2.  Find  a  fourth  proportional  to  3,  4,  and  2. 

3.  Find  a  mean  proportional  to  2;^  and  9^. 

4.  Find  a  third  proportional  to  2^  and  OJ. 

5.  18,  X,  and  50  are  in  continued  proportion;  find  a 
fourth  proportional  to  them,  eliminating  x. 

6.  Find  a  third  proportional  to  a^  and  Zd^h]  to  ^aW  and 
a?\  and  of  these  two  results  find  the  geometric  mean. 

7.  Of  several  numbers  in  continued  proportion,  if  the 
second  and  third  are  4  and  6,  what  is  the  fifth  ? 

8.  Find  a  number  to  which  if  2,  12,  and  17  be  sepa- 
rately added,  the  three  results  are  in  continued  proportion. 

9.  Of  two  numbers  one  is  7  greater  and  the  other  3  less 
than  their  mean  proportional.     What  are  the  numbers  ? 

10.  Of  four  numbers  in  continued  proportion,  8  is  the 
least  and  27  the  greatest.     Find  the  other  two. 

VARIATION. 

332.  In  any  algebraic  investigation  there  may  be  two 
kinds  of  quantities:  variables,  whose  values  are  subject  to 


306  ALGEBRA, 

change,  either  continuously  or  at  intervals ;  and  constants, 
whose  values  are  understood  to  be  fixed  and  unchangeable 
throughout  the  investigation. 
Thus  in  the  expression 

x^  -Sx-\-  15 
x'  -  7^;  +  12 

X  may  assume  any  value  we  choose  to   impose  upon   it; 
while  8,  15,  7,  and  12  are  constants. 

333.  The  relation  between  the  variables  x  and  y  in  the 
equation 

X  =  ley     {when  the  constant  h  is  unknown) 

is  sometimes  expressed  x  cc  y;  to  be  read  ^^  x  varies  as  ^." 
A  value  of  x  and  a  value  of  y  which  when  substituted 
together  in  this  equation  make  it  a  true  statement  are 
called  simultaneous  values,  or  corresponding  values,  of  x 
and  y.  They  may  be  represented  with  equal  suffixes, thus: 
x^  and  y^,  x.^  and  y^^  x^  and  y^  are  simultaneous  pairs,  and 
we  have  the  equations 

^,  =  ^y.\      J  =  >^ 

Hence     —  =  —  ;     or  by  alternation     -  =  — . 

y.     y.  ^,    ^. 

334.  If  one  quantity  varies  as  another,  a  proportion 
may  be  formed  by  using,  for  homologous  terms  of  the  pro- 
portion, corresponding  values  of  the  variables. 

Model  A. — The  weight  of  wire  of  a  certain  kind  varies 
as  its  length;  if  1  metre  of  it  weighs  .8  grams,  how  much 
will  2.35  metres  weigh  ? 


PROPORTION  AND    VARIATION  807 

Here/,  =  1;  w,  =  .8;  /,  =  2.35. 

1  :  .8  =  2.35  :  w^ 
^^  =  .8  X  2.35  ~  1.88  grams.     Ans, 

EXERCISE   CLIX. 

1.  li  X  varies  as  y,  and  x  =  ^  when  y  =  10,  find  the 
value  of  X  when  ^  =  11. 

2.  U  X  oc  a,  and  x  =  1.09  when  y  =  .7,  find  y  when 
a:  =  23. 

3.  If  X  ex  z,  and  z  =  p  when  x=  q,  find  :?;  when  z  =  q, 

4.  If  a  a  Z>,  and  «^  =  5'''^  —  j.;^  when  I)  =  §''^,  find  Z>  when 
a=pq{p-q),  _ 

5.  If  h  a  ^,  and  ^  =  1^3  when  h  =■  Vb,  find  h  when 

335.  One  variable  may  vary  as  the  reciprocal  of  the 
other,  or  as  its  square  or  cube,  'or  as  any  algebraic  expres- 
sion involving  the  other  variable. 

The  proportion  will  then  be  constructed  by  taking  for 
homologous  terms  the  same  algebraic  expression  with  the 
different  values  of  the  variable.     Thus  \i  y  <x  x^  -\- '^x  —  b 

y^  :  x^^  +  3a;,  -  5  =  ^, :  x^^  +  3a;,  -  5. 

Or,  if  y"^  -\-  2y^  ca  x^  —  x 

y^  +  2y,3  :  x^  -x^=^  y,^  +  2y,^  :  x^^  -  x^. 

336.  When  one  variable  varies  as  the  reciprocal  of  an- 
other they  are  said  to  vary  inversely;  in  distinction  from 
this,  the  case  of  variation  first  described  is  called  direct 
variation. 


308  ALGEBRA. 

Thus  when  x  varies  inversely  as  ^^,  a;  a  ^ ;  when  x  varies 
directly  as  y^,  x  oc  y^. 

Often  the  expression  "varies  inversely  as"  is  replaced  by  tlie 
words  •'  is  reciprocally  proportional  to  "  ;  and  "  varies  directly  as" 
may  be  written  "is  directly  proportional  to,"  or  simply  "is  propor- 
tional to." 

EXERCISE    CLX. 

1.  If  an  animal's  strength  is  proportional  to  the  square 
of  his  length,  and  an  animal  4  feet  long  can  pull  1250 
pounds,  how  much  can  an  animal  4.3  feet  long  pull  ? 

2.  If  the  weight  of  a  body  is  reciprocally  proportional  to 
the  square  of  its  distance  from  the  centre  of  the  earth,  how 
much  weight  will  a  3-pound  ball  indicate  on  a  spring  balance, 
at  a  distance  above  the  earth  equal  to  the  diameter  of  the 
earth  ? 

d,  \ix  varies  inversely  as  y'^  +  y,  and  x=  b  when  y  ~2, 
what  will  x  equal  when  ^^  =  3  ? 

4.  The  area  of  an  equilateral  triangle  varies  as  the  square 
of  its  side;  and  an  equilateral  triangle  whose  side  is  5  feet 
long  incloses  about  10.825  square  feet.  Find  the  area  of 
an  equilateral  triangle  whose  side  is  8  feet  long. 

5.  If  the  intensity  of  light  varies  as  the  square  of  its  dis- 
tance, and  if  a  certain  lamp  at  a  distance  of  10  feet  gives 

an  intensity  —  that  of  sunlight,  what  will  be  its  intensity 

at  a  distance  of  50  feet  ? 

337.  If  there  are  three  quantities  so  related  that  the  first 
varies  as  the  second  when  the  third  is  constant,  and  varies 
as  the  third  when  the  second  is  constant,  then  the  first 
varies  as  the  product  of  the  second  and  third  when  they 
both  vary. 

Proof,  Let  x^y^z^,  x^y^z^  be  corresponding  values  of  the 
three  variables.     If  we  suppose  y^  to  be  constant,  and  z  to 


PROPORTION  AND    VARIATION.  309 

vary  from  z^  to  z^,  we  shall  have  a  new  value  of  x,  which 
we  may  call  a;',  for  which  this  equation  holds: 

©-;  =  '- 

^  x'      z^ 

Again,  if  we  suppose  z^  to  be  constant,  while  y  varies  from 
Vi  to  y^,  we  shall  have  the  value  of  x  corresponding  to  y^ 
and  z^,  that  is,  x^\  and  for  that  this  equation  holds  : 

^^  ^'      Vi 
®  -=- 

Then  by  multiplying  Q  and  @  we  obtain 

and  as  these  are  any  two  sets  of  corresponding  values,  the 
proof  is  general. 

This  theorem  is  of  great  importance  in  geometry. 


CHAPTER  XIII. 

THE  PROGRESSIONS. 

338.  A  succession  of  algebraic  terms  which  progress  ac- 
cording to  some  definite  law  is  called  a  series. 

339.  Among  the  simplest  of  series  is  that  in  which  the 
terms  progress  by  a  common  difference;  thus 

7;  10;  13;  16;  19;  22;  25;  etc. 
5x  —  dy;  6x  —  y;  6x  -{-  y;  6x  +  Sy;  6x  +  5y;  etc. 

In  the  first  illustration  above,  the  common  difference  is  8;  the  first 
term  is  7;  3  is  added  to  7  once  to  give  the  second  term,  twice  to  give 
the  third  term,  3  times  to  give  the  fourth  term,  and  so  on. 

In  the  second  illustration  the  first  term  of  the  series  is  6x  --  Sy; 
the  common  difference  is  2y. 

ARITHMETIC   PROGRESSION. 

340.  A  series  in  which  the  terms  progress  by  a  common 
difference  is  called  an  Arithmetic  Progression.* 

In  deducing  formulae  for  the  laws  of  such  a  series,  the 
letter  a  represents  the  first  term,  d  the  common  difference, 
s  the  sum  of  n  terms,  I  the  ^th  term. 

The  first  formula  is  seen  at  once  : 

(i)     l  =  a+{n-  l)d. 

*  Usually  abbreviated  A.  P. ;  pronounced  arithmetic  progression. 

310 


THE  PROGRESSIONS,  311 

The  formula  for  the  sum  of  n  terms  is  obtained  by  first 
observing  that  the  series  can  be  written  in  reverse  order  by 
beginning  with  I  and  subtracting  d,  then  for  the  sum  of  the 
series  we  have  two  expressions  : 

s  =  a+{a  +  d)  +  {a+'id)  -{-  (a  +  3^)+ (a  + 4^)  ,  ,  ,  I 
s  =  l-\.{l-^d)-\-{l-^d)  +  {l-M)  +  {l''4td)  ...  a 

Adding  these  two  we  have 

<;ts  =  {a+l)  +  [a  +  l)  +{a  -\-l)  +{a  +  l)  .  .  .  +{l  +  a) 
=  n{a  +  I) 

(II)     s  =  l{a+l) 

341.  These  two  formulae  contain  all  the  independent 
conditions  that  are  imposed  by  the  definition  on  the  five 
unknown  letters  a,  d,  I,  ii,  s.  Hence  we  can  determine 
any  one  of  them,  only  if  the  three  others  are  given. 

Finding  I  and  s. 

342.  Model  A. — Find  the  10th  term  in  the  series 

35,  31,  27  .  .  . 
a  =  35;  d  =  —  4:;  n  =  10 
Z  =  35  +  9(-  4)  =  -  1 

Find  the  sum  of  the  series,  supposing  it  stops  with  the 
13th  term. 

The  13th  term  is  Z  =  35  +  12(-  4)  =  -  13 
s  =  -V-(35  -  13)  =  13(22)  =  286 

EXERCISE    CLXI. 

1.  11;  13;  15;  .   .   .;  100th  term. 

2.  38;  30;  22;  .   .   .;  18th  term. 

3.  -100;   -91;   -82;  .   .   .;  50th  term. 


312  ALGEBRA, 

4.  —  100;  —  91;   —  82;  .   .   .;  sum  of  15  terms,  be- 
ginning v/ith  the  3d  term. 

5.  83;  87;  91;  .  .  .;  to  20  terms;  sum  of  the  series. 
6-  i;  i;  i;  .  .  .;  lOthterm. 

7.  ^V;  i;  ^V;  •   •  •;  28th  term. 

8.  .001;  .0055;  .01;  .  .   .;  23d  term. 

9.  Sum  of  the  first  ten  terms  in  ^V>  i?   sV?  •  •  • 

10.  Find  the  last  term  and  the  sum  of  the  terms  in  an 
A.  P.  of  11  terms  of  which  the  first  two  are  a  —  x  and 
a-y. 

Eliminating  with  the  Formulae. 

343.  If  any  three  of  the  five  constants  of  an  A.  P.  be 
given,  substituting  in  the  formulae  (I)  and  (11)  gives  two 
equations  with  two  unknown  letters,  which  can  be  deter- 
mined by  elimination. 

Model  B.— Given  ^  =  16;  ^  =  4;  5  =  88; 
find  n. 
©  Z  =:  16  +  (w  -  1)4  =  12  +  4/1 

©  5  =  88  =  |(16  +  1) 

@  88  =  -(28  +  4^)  subst.  ©in  © 

®  4^^2  +  287i  =  176  ®  X  2 

®  7i2  +  77i  -  44  =  0  ®  -4-  4  -  44 

®  (7i  +  ll)(7i-4)  =  0 
7^  =  4;  n  =  —11 

Of  these  answers  ^  =  4  is  the  only  one  that  satisfies  the 
conditions  of  the  problem. 


THE  PROGRESSIONS. 


313 


EXERCISE    CLXII. 

In  the  following  examples  find  the  constants  that  are  not 
given : 


a 

d 

I 

n 

s 

1. 

5 

^  10 

2 
3 

25 

2. 

38 

3 

8 
-3 

5 

1572 

4. 

787 

80 

5. 

6 



-34 

-294 

6. 

11 

....  .... 



36 

-2754 

7. 

4 

57 

16 

8. 

-7 

-36 

-92 

9. 

397 

200 

19700 

10. 

11 

13 

182 

11 

11 

22 

231 

12. 
13. 
14. 
15. 

I 

7 

9 

1- 

100 

"'9771* 
*  24* 

37 
-  6 

11 

16 

if 

5 

17 

3 

17. 
18. 
19. 
20. 

tV 

25^ 

f 
-9f 

21 
25 

-  993 

-1 

-  181i^ 

21. 

i 

n 



33| 

EXERCISE    CLXIII. 

Determine  formulm  for  each  constant  in  terms  of  three 
others,  as  follotvs : 

1.  Find  5;  given  a,  d^  and  L 

given  a,  d,  and  n, 
[given  a,  I,  and  n ;  formula  11]. 
given  d,  I,  and  n. 

2.  FindTi;  given  a,  d,  and  I. 

given  a,  d,  and  s. 
given  a,  I,  and  s. 
given  d,  I,  and  s. 


314  ALOEBBA. 

3.  Find  Z;  [given  a,  rZ,  and  7i]  formula  i]. 

given  a,  d,  and  s. 
given  a,  n,  and  s, 
given  <i,  ^,  and  s. 

4.  Find  (i;  given  a,  I,  and  ^. 

given  a,  I,  and  5. 
given  a,  n,  and  5. 
given  Z,  ^,  and  5. 

5.  Find  a;  given  (^,  I,  and  ^^. 

given  6?,  I,  and  5. 
given  ^,  n,  and  5. 
given  Z,  71,  and  5. 

If  one  liad  a  great  many  examples  to  work  under  any  one  of  the 
20  cases  just  given,  it  would  pay  to  use  one  of  the  formulae  specially 
derived  for  that  case.  But  in  solving  such  as  occur  occasionally  it 
is  better  to  use  the  formulae  (i)  and  (ii),  substituting  the  given  con- 
stants and  deriving  the  others  by  elimination. 

EXERCISE   CLXIV. 

In  Arithmetic  Progression^ — 

1.  If  the  first  term  is  3  and  the  5th  term  is  67,  how 
many  terms  must  be  taken  to  add  up  179  ? 

2.  When  the  10th  term  is  6  and  the  15th  term  is  —  2, 
find  the  sum  of  the  first  8  terms. 

3.  When  the  5th  term  is  10  and  the  13th  term  is  101, 
find  the  sum  of  alternate  terms,  beginning  with  the  3d 
and  ending  with  the  13th. 

4.  When  the  sum  of  the  first  7  terms  is  equal  to  3  times 
the  next  term,  and  the  term  after  that  is  —  2,  what  are  the 
first  3  terms  of  the  series  ? 

5.  When  the  sum  of  the  first  3  terms  is  equal  to  half  the 
sum  of  the  first  7  terms,  and  greater  by  14  than  the  sum 
of  the  second  group  of  seven  terms,  what  is  the  16th 
term  ? 


THE  PROGRESSIONS.  315 


GEOMETRIC   PROGRESSION. 

344.  A  series  in  which  the  terms  progress  by  a  common 
ratio  is  called  a  Geometric  Progression.* 
Such  are  the  series, — 


11; 

32; 

44; 

88; 

176;  . 

24; 

8; 

3|; 

■ff. 

A;. 

x^; 

x'y; 

xy'^; 

y^; 

In  each  of  these  series  the  ratio  of  each  term  to  the  term  preceding 
is  the  same  throughout  the  series.  The  first  term  of  the  first  series 
is  multiplied  by  2  to  give  the  second  term,  by  2^  to  give  the  third 
term,  by  2^  to  give  the  fourth  term,  and  so  on. 

345.  In  deducing  formulae  for  G.  P.,  we  use  the  same 
letters  as  in  A.  P.,  except  that  instead  of  d  for  a  common 
difference  we  have  r  for  a  common  ratio. 

The  first  formula  for  Gr.  P.  is  seen  at  once : 

(in)     I  =  ar""-^ 
For  the  sum  of  n  terms  we  have 

s  =  a  -\-  ar  -{-  ar^  -\-  ar^  -\-  ar^  -\-  ar^  -\-  .  .  .  -{-  ar^~^ 
=  a{l  +  r  +  r^  +  r^  +  r^  +  r^  ■+-...  +  r^-i) 

The  factor  in  parenthesis  will  be  recognized  as  the 
quotient  of  (1  —  r")  -t-  (1  —  r) ;  hence 

1  -  r"         r^  -  1 

(iv)     s  =  a- =  a — — - 

^     ^  1  —  r  r  —  I 

Since  ar""  =  {ar''~'^)r  =  Ir,  we  may  write 

Ir  —  a 


(lY) 


r-1 

Usually  abbreviated  G.  F, 


316  ALGEBRA, 

Finding  I  and  s. 
346.    Model   C.  —  Find  the   10th.  term  of  the  series 
~;  x^',  x'y',  .  .  . 

^^  y  ^r. 

a  —  -  \     r  —  -\     n  —  Vd 

y  ^ 


i^(A(yy^tf^ylAns. 

\y  /  \x  /       x^y      x"" 


^9        ,,8 

\y  J  \x  1       x^y 

Model  D. — Find  the  sum  of  the  first  six  terms  of  the 
G.P.f;!;... 


=  ii4)l_i  =  02. (1)5  _  II  ^  i,||8  jins. 


bT^  =  -V-{  (1)^-1}  -Hfl 


EXERCISE    CLXV. 

1.  2560;  1280;  640;  .  .  .;  11th  term. 

2.  486;  162;  54;  .  .  .;  sum  of  first  six  terms. 

3.  tV;  i;  1^  •  •  •;  8th  term. 

4.  1 ;  2 ;  4 ;  .  .  . ;  sum  of  first  seven  terms. 
6.  .0003;  .003;  .03;  .  .  .;  10th  term. 

6.  3;   —  6;  12;  —  24;  .  .  .;  sum  of  first  eleven  terms. 

7.  ic;  1;  — ;  .  .  .;  20th  term;  Mh  term. 

X 

8.  5;   —  ^;   +  gV;  .  .  . ;  sum  of  first  five  terms. 

9.  x^\  x^y\  x'^y'^]  .  .  .;  sum  of  first  five  terms. 

/» /v 

10.  a^  —  x^i  a  —  x\  — ; — ;  .  .  .;  sum    of    first    seven 
'  a-\-  X 

terms. 


THE  PROGRESSIONS. 


317 


Eliminating  with  the  Formulae. 

347.  If  any  three  of  the  five  constants  of  a  G.  P.  be 
given,  substituting  in  the  formulas  (iii)  and  (iv)  gives  two 
equations  with  two  unknown  letters,  which  can,  theoretically 
at  least,  be  determined  by  elimination,  as  in  A.  P. 

As  a  matter  of  fact  there  are  twelve  cases  where  a  solution 
is  possible,  and  in  these  cases  elimination  is  not  necessary; 
in  the  other  eight  cases  the  equations  obtained  by  substitu- 
tion require  either  the  use  of  logarithms  or  the  solution  of 
higher  equations. 


EXERCISE    CLXVI. 

In  the  following  examples  find  the  co7istants  that  are  not 
given : 


a 

r 

I 

n 

s 

1. 

1215 

4 

6 

2. 

64 

-i 

1 

8. 

2 

10 

7161 

4. 

5 

625 

8 

5. 

* 

7 

95i 

6. 

f 

h 

.05155 

7. 

2 

1 

5 

8 

2 
5 

5 

781250 
160 

9. 

6 

10. 

-i 

-^ 

6 

11. 

5 

327680 

9 

12. 

4i 

M 



6 

13 

-5 
.5 

2 
3 

y 

X 

425 

14 

546.5 

15. 

/ 

8 

318  ALGEBRA. 

Inserting  Means. 

348.  The  terms  which  intervene  between  the  first  and 
last  terms  in  a  finite  series  are  called  means.  If  the  series 
is  in  A.  P.  these  are  called  arithmetic  means;  if  in  G.  P., 
geometric  means. 

When  there  is  only  one  mean,  it  is  known  as  the 
arithmetic  mean  (or  the  geometric  mean,  as  the  case  may 
be),  of  the  two  quantities  that  serve  as  extremes. 

When  there  are  two  or  more  means,  they  with  the 
extremes  make  a  finite  series,  in  which  the  number  of 
terms  is  two  more  than  the  number  of  means. 

Model  E. — Insert  5  geometric  means  between  16  and  182|-. 

a  =  16;  1=  182J;  n  =  l 
182^  =  16r6 

-w-  =  ^ 

Series  is  16,  24,  36,  54,  81,  ^^y  ^K 

Ans.  24;  36;  54;  81;  121J. 

EXERCISE    CLXVII. 

1.  Insert  6  arithmetic  means  between  3  and  10. 

2.  Insert  6  arithmetic  means  between  3  and  13. 

3.  Insert  5  arithmetic  means  between  3  and  10. 

4.  Insert  7  arithmetic  means  between  3  and  10. 

5.  Find  the  arithmetic  and  geometric  means  between  15 
and  900. 

6.  Insert  3  geometric  means  between  15  and  240. 

7.  Insert  3  geometric  means  between  15  and  1215. 

8.  Insert  4  geometric  means  between  ^  and  512. 

9.  Insert  6  geometric  means  between  14  and  1792. 

10.  Find  the  arithmetic  and  geometric  means  of  the 
numbers  3^  and  27723^. 


THE  PM0QEE88I0N8,  319 

11.  Insert  101  arithmetic  means  between  —  2  and  17. 

12.  Insert  9  arithmetic  means  between  a  —  h  and  a-\-i, 

13.  Insert  4  geometric  means  between  x^  and  32?/^^. 

14.  Insert  10  arithmetic  means  between  J^-  and  10. 

15.  Insert  x  arithmetic  means  between  —  and  x, 

X 

16.  Insert  I  arithmetic  means  between  a  —  h  and  a-^l. 

17.  Insert  x  —  \  geometric  means  between  x  and  ax, 

4        -.34/3" 

18.  Insert  4  geometric  means  between  —  and  — ^ — 

19.  Find  the  arithmetic  mean  and  the  geometric  mean 
between  a^  —  da  —  2  and  a^  —  4:a^  +  5a  —  2. 

a* 

20.  Insert  7  geometric  means  between  16a"^  and  — . 


Infinite  Geometric  Series. 

349.  When  in  a  geometric  progression  the  value  of  r  is 
less  than  1,  the  values  of  the  successive  terms  become  less 
and  less,  and  the  value  of  I,  the  ^th  term,  may  be  made  as 
small  as  we  choose  by  taking  n,  the  number  of  terms,  large 
enough.     In  the  formula 

,     .  Ir  —  a 

(iv)   s  = — 

V     /  r  —  1 

the  term  Ir  may  be  made  smaller  than  any  quantity  that 
can  be  assigned,  if  we  are  permitted  to  take  a  sufficient 
number  of  terms;  in  other  words,  if  the  number  of  terms 
is  unlimited,  the  more  terms  we  take  the  nearer  s  comes  to 
the  value 


-        —  a 


r  —  1       1  —  r 


320 


ALOEBBA, 


This  is  what  is  meant  by  the  statement  : 
If  the  number  of  terms  in  a  G.  P.  is  infinite,  and  r  is  less 
than  1,  then  the  sum  of  the  series  is 


350.  This  result  may  be  confirmed  by  long  division,  from 
which  we  obtain 

n 

=  a  -\-  ar  +  ar^  +  ar^  +  ar^  -{-  ar^  +  ar^  -{-  .  ,  , 


1  -  r 


the  number  of  terms  being  continued  without  limit. 

351.  Now  this  last  identity  can  be  tested  with  different 
values  of  r.     Let  us  try  r  =  2. 


1-2 


[=—a\=a  +  2a  +  4a  +  8«+16«+32a  +  64a+  . , 


The  more  terms  we  take  of  the  infinite  series  on  the  right- 
hand  side  of  this  equation,  the  more  their  sum  diverges 
from  the  value  of  the  expression  that  gave  rise  to  the  series. 
We  come  to  this  same  curious  conclusion  whenever  r  >  1. 

Again,  suppose  r  =  -J. 

n=^L^Tj~^"^'3+"9  "^27  +  8l"^243"^729"''  *  *  * 

The  successive  values  of  5,  for  different  values  of  n,  are 
shown  in  the  following  table  : 


Value  of  n 

1 

2 

3 

4 

5 

6 

Value  of  5 

a 

4a 
3 

13a 
9 

40a 

27 

121a 
81 

364« 

243 

Difference  between  s  and  — . 

a 

a 

"6 

a 

18 

a 
54 

a 
162 

a 

486 

tb:b}  progressions.  321 

It  is  evident,  then,  that  in  this  case,  where  r  =  i,  the 

more  terms  we  take,  the  closer  their  sum  comes  to  the 

,      2a 
value  —. 

352.  An  infinite  series  in  which  the  sum  of  the  first  n 
terms  approaches  a  definite  limit  as  n  is  indefinitely  in- 
creased is  called  a  convergent  series;  if  the  sum  of  the 
first  n  terms  does  not  approach  a  definite  limit  as  n  is 
indefinitely  increased,  the  series  is  said  to  be  divergent. 
This  distinction  is  of  the  utmost  importance  in  the  further 
study  of  algebra. 

Model  F. — Find  the  sum  of  the  infinite  series 

8,     4,     2,  .  .  . 

n  « 

=  16 


l-r      l~i 
Model  G. — Find  the  value  of  the  circulating  decimal 
3.7182* 
3.7i82  =  3.7  +  .Oi82 
.0i82  =:  .0182  +  .0000182  +  .0000000182  +  etc. 

_  182       182      182 

~  10*  "^  10^  "^  10^0  "*"  •  • 


Here  a 

182  _ 
~10*' 

r  = 

103-     Hence 

«' 

182 
~  10*  ' 

^(- 

1  \  _ .0182  _ 
10«y  ~  .999 

182 
9990 

Then 

3.7 

=  nm- 

3.7  +  , 

.0182 

■  =  mu  +  ifWo 

=  3i 

*  This  is  an  abbreviation  for  8.7182182182182,  where  the  repeti- 
tions of  182  are  continued  indefinitely. 


322  ALQJEIBBA. 


EXERCISE  CLXVIII. 


Find,  wherever  possible,  the  sum-  of  the  infinite  series: 

1.  54;  18;  6;  .  .  .  6.  1;  .2;  .04;  .  .  . 

2.  .7;  .07;  .007;  ...       7.  2.7;  .9;  .3;  .  .  . 

3.  I;  i;  ^V;  •  •  .  8.  1.728;  1.44;  1.2;  .  .  . 

4.  I;  I;  1;  .   .   .  9.  7;  4.9;  3.43;  .  .   . 
6.  f;  1;  I;  .   .  .              lo.  130;  26;  5.2;  .  .  . 

Find  the  values  of  the  circulating  decimals  : 

11.  5.7.  16.  3.736. 

12.  5.73.  17.  8.18. 

13.  5.735.  18.  10.201. 

14.  5.73.  19.  702. 

15.  1.01101101101101  ...  20.  100.8. 

21.  The  sum  of  an  infinite  series  in  G.  P.  is  33^  per 
cent  greater  than  the  sum  of  the  first  two  terms.  What 
must  be  the  value  of  r  ? 

22.  The  sum  of  the  fifth  and  sixth  terms  of  a  series  in 
G.  P.  is  12  times  the  sum  of  the  second  and  third  terms. 
What  is  the  ratio  of  the  seventh  term  to  the  first  ? 

23.  The  geometric  mean  of  the  first  and  fifth  terms  of  a 
geometric  series  is  100;  if  the  second  term  is  20,  what  is 
the  fifth  term  ? 

24.  The  geometric  mean  of  the  second  and  sixth  terms 
of  a  G.  P.  is  500;  that  of  the  fifth  and  seventh  terms  is  20. 
Find  the  series. 

25.  In  a  G.  P.  the  arithmetic  mean  of  the  first  two 
terms  is  5  times  the  sum  of  the  second  and  third  terms. 
Find  r. 

26.  Three  numbers  whose  sum  is  38  are  in  G.  P. ;  if  1, 
2,  and  1  are  added  to  them,  in  order,  the  results  will  be  in 
A.  P.     Find  the  numbers. 

27.  In  a  certain  G.  P.  the  ratio  of  the  third  term  to  the 


THE  PR00RBS8I0N8.  323 

fifth  is  7;  what  is  the  ratio  of  the  first  term  to  the  seventh 
term  ? 

28.  How  many  terms  of  the  series  37,  33,  29,  .  .  . 
amount  to  187  ?     Explain  the  two  answers. 

29.  By  how  much  is  the  sum  of  the  infinite  series  256, 
64,  16  .  .  .  diminished  if  all  terms  after  the  fifth  are  cut 
off? 

30.  In  finding  the  sum  of  the  infinite  series  9 ;  — ■  3 ;  1 ; 
.  .   .  what  is  the  error  if  we  stop  with  the  10th  term  ? 

31.  If  in  a  given  G.  P.  we  find  r  =  .1,  find  r  for  the 
series  obtained  by  taking  every  fourth  term  of  the  given 
series. 

32.  Find  an  infinite  series  in  G.  P.  such  that  each  term 
is  4  times  the  sum  of  all  the  terms  that  follow  it. 


HARMONIC  PROGRESSION. 

353.  A  series  in  which  the  reciprocals  of  the  terms  are 
in  arithmetic  progression  is  called  a  Harmonic  Progres- 
sion.* 

Thus  the  numbers 

60;    30;    20;    15;    12;    10 
are  in  H.  P.  because  their  reciprocals 

•bV)    -to?    "O"^    "/o"?    to  5    /o" 

are  in  A.  P. 

354.  Problems  in  H.  P.  are  solved  by  converting  the 
given  constants  into  the  corresponding  constants  of  arith- 
metic progression. 

*  Usually  abbreviated  H.  P. 


324  ALGEBRA, 

Model  H.— Insert  3  harmonic  means  between  6  and  30. 
This  corresponds  to  an  arithmetic  progression  in  which 

i  +  4^  =  ^V 
5  +  1206?  =1 
120^  ==44 

^  =  -  sV 
In  A.  P.:  i;  ,-\;  -^-V;  ^V;  sV 
InH.  P.;  6;  7^;  10;  15;  30. 

EXERCISE    CLXIX. 

1.  What  is  the  fifth  term  of  a  H.  P.  of  which  the  second 
term  is  5  and  the  fourth  term  is  10  ? 

2.  Find  the  sum  of  the  first  five  terms  of  a  H.  P.  of 
which  the  second  and  third  terms  are  ^  and  \, 

3.  Write  down  the  fifth  and  sixth  terms  of  a  H.  P.  of 
which  the  second  term  is  1  and  the  third  term  2. 

4.  The  sum  of  three  terms  of  a  H.  P.  is  If  and  the  last 
term  is  |.     What  are  the  other  terms  ? 

5.  Insert  four  harmonic  means  between  12^  and  75. 

6.  The  first  term  of  a  harmonic  series  is  15015  and  the 
fourth  term  is  5005.     What  is  the  seventh  term  ? 

7.  Insert  four  harmonic  means  between  220  and  1540. 

8.  The  first  two  terms  of  a  harmonic  series  are  —  3  and 
+  3;  find  the  third  term. 

9.  Insert  four  harmonic  means  between  —  3640  and  455. 

10.  Insert  five  harmonic  means  between  1  and  8;  five 
geometric  means;  five  arithmetic  means. 

The  Three  Means. 

355.  The  harmonic  mean  of  any  two  quantities  is  repre- 
sented by  H\  the  geometric  mean  by  G',  and  the  arithmetic 


THE  PROGRESSIONS.  325 

mean  by  A.     They  are  determined  by  formulae  derived  as 
follows,  in  each  case  from  the  definition  of  the  series.     Let 
X  and  y  represent  the  extremes. 
A]  by  definition  oi  A.,V.,  A  —  x  =  y  —  A 

%A=:x  +  y  (T)  j^x  +  A 

G'y  by  definition  of  G.  P.,  G  :x  =  y  :  0 

xy  =  G^  (j)  X  xG 

G  =  Vxy 

H\  by  definition  of  H.  P.,   -— = —r 

^     ^  H      X       y       H 

H-x'^y-     xy  ^  ^  x  '^  H 

H  =  -^  1:®X2 

356.  For  distinguishing  among  the  three  types  of  series 
investigated  in  this  chapter,  the  preceding  formulae  furnish 
tests.     Thus  the  three  terms 

60;         70;         84 

60  +  84 


are  not  in  A.  P., — because  70  :^ 


2 


and  not  in  G.  P., — because  70  ^  V60  x  84 

1.  ^  4-1.  •     XT   r»       1.  ry^       2  X  60  X  84 

but  they  are  m  H.  P., — because  70  =  — ^^   ,   ^, — 
•^  60  +  84 


326  ALGEBRA. 


EXERCISE  CLXX. 


1.  Prove  by  this  test  that  the  three  means  A,  0^  H,  for 
AKY  two  numbers  always  form  a  G.  P. 

2.  Prove  that  in  the  proportion  x  :  y  =p  —  x  :  y  —  p, 
p  is  Si  harmonic  mean  bafcween  x  and  «/.* 

Identify  the  following  series  as  A,  P.,  G.P.,  or  H.  P.: 

3.  i;  4;  i>  •  •  •      4.  J;  i;  i;  . . .      6.  i;  i;  f ;  •  •  • 

6.  3;  6;  9;  .  .  .       7.  4;  2;  0;  .  .  .       8.  7^;  10;  15;  .  .  . 

9.  34;  5;  10;...    lo.  2^;  5;  10;  .  .  .    il.  65;105;  273; .. . 

V2         ,—       ,-  V6  —  2     1      VE+2 

12.  3—;   iVdO;   V2;  ...    13.  -^— ;  3-;   —3^  5  ... 

14.1(^8  +  5);  ^'<''\+  %  U^S-S);   ... 

16.  275;    385;    539;   ...  I6.  924;    1309;    2244;    ... 

17.  4788;    3800;    3150;    .  .  . 

18.  Insert  7i  arithmetic  means  between  a  and  b  and  find 
the  sum  of  the  series  thus  formed. 

19.  If  a  -\-  b,  b  -}-  c,  and  c  -\-  a  are  in  G.  P.,  prove  that 
a  -}-  b  _c  ~  a 

b  -{-  c~  a  —  b' 

20.  If  y  —  x-\-z  —  x;  x  —  y;  1/  are  in  G.  P.,  prove  that 
y;  x;  z  also  form  a  G.  P. 

21.  There  are  m  arithmetic  means  between  2  and  47, 
and  the  ratio  of  the  4th  to  the  {m  —  3)**^  is  |.     Find  m, 

22.  Find  a  number  to  which  if  2,  8,  and  17  be  added 
the  results  are  in  G.  P. 

23.  The  arithmetic  mean  of  two  numbers  exceeds  the 
geometric  by  24,  and  the  geometric  exceeds  the  harmonic 
by  30.     Find  the  numbers. 

*  This  property  is  sometimes  used  to  define  H.  P. 


THE  PROGRESSIONS,  327 

24.  The  arithmetic  mean  of  two  numbers  exceeds  the 
geometric  by  a,  and  the  geometric  exceeds  the  harmonic 
by  h.     Find  the  arithmetic  mean  of  the  numbers. 

25.  Given  A  and  G  for  any  two  numbers,  find  a  formula 
for  the  numbers. 

26.  Given  A  and  iZ"for  any  two  numbers,  find  a  formula 
for  the  numbers. 

27.  Given  H  and  G  for  any  two  numbers,  find  a  formula 
for  the  numbers. 

28.  Find  an  A.  P.  whose  first  term  is  7  and  whose  sec- 
ond, fifth,  and  tenth  terms  form  a  G.  P. 


CHAPTEK   XIV. 
DETACHED   COEFFICIENTS  ;   BINOMIAL   THEOREM. 

Detached  Coefficients  in  Multiplication. 

357.  In  multiplying  expressions  which  contain  only  one 
letter,  after  the  expressions  are  arranged  by  descending 
powers  of  that  letter,  each  term  in  the  product,  as  in  the 
multiplicand  and  multiplier,  is  determined  as  to  its  degree 
by  its  position  in  the  expression.  This  fact  leads  to  a  very 
convenient  method  in  Algebra,  called  the  method  of  de- 
tached coefficients,  which  has  several  very  important  appli- 
cations. 

Model  A.  — Arranging  similar  terms  in  columns : 
(3a;3  -  2a;2  +  5a;  -  3)(2a;2  +  3a;  -  1). 
3^3  _  2a;2  +    bx  -    3 
2a;2+    3a;  -    1 
6a;5  -  4a;4  +  lOa;^  -    6a;2 

+  9a;4-    6a;3+15a;2-    9a; 

-    3a;^+    2a;2-    5a;  +  3 
6a;5  +  bx^  +      x^  +  lla;^  -  14a;  +  3 
The  entire  significance  of  this  operation  is  preserved  if 
we  write  only  the  coefiicients,  thus : 
3__2+    5-    3 
2+    3-    1 
6-4  +  10-    6 

9-    6  +  15-    9 
-    3+    2-    5  +  3 
6  +  5+    1  +  11-14  +  3 


DETACHED  COEFFICIENTS,  329 

In  the  same  way  the  multiplication  (5^^  +  ^x^  —  2ic  +  3) 
{^x^  —  2a;  +  3)  may  be  written: 
Model  B. 

5+    3-    2+    2 
3-2+3 


15+9-6+6 

_  10-    6+    4-    4 

15+    9-    6  +  6 
15-    1+    3  +  19-10  +  6 
The  answer  being  Ibx^  —  x^ -\-  Zx^-\- 19^^  _  ^q^  _|_  g^ 

In  the  following  multiplication  notice  the  significance  of 
the  zero  coefficient  in  the  result. 

Model  C. 

{^x^  +  bx^  -  2x^  +  7x-  2){5x^  +  2x^  -  7a;  -  2). 
3+5-2+7-2 

5  +  2-7-2 

15  +  25-10  +  35-10 

6  +  10-4  +  14-4 
-21-35  +  14-49  +  14 

-    6-10+    4-14  +  4 

15  +  31-21-10+    8-49+    0  +  4 

The  answer  being  written  : 

152:'^  +  dlx^  -  21a;5  -  lOo;*  +  Sx^  -  4t9x^  +  4 
Model  D.     {3x^  +  5)(2:^;3  _|_  ^  _  7). 
3  +  0  +  5 
2+0+1-7 
3  +  0  +  5 

3+    0  +  5 
-21  +  0-35 
3  +  0  +  8-21+5-35 
3x^  +  Sx^  -  21^2  +5^-35 


330  ALGEBRA, 

The  ordinary  notation  of  numbers  is  by  detached  coefficients  ;  thus 
327  is  really  the  same  as  W^  +  2^  +  7  where  ^  =  10;  and  5002546  is  the 
same  as  5^*  +  2^*  +  5^^  +  4^  +  6.  It  is  complicated,  however,  by 
the  fact  that  whenever  the  coefficient  rises  to  the  value  of  i  the  term 
becomes  of  higher  degree. 

EXERCISE  CLXXI. 

By  detached  coefficients  multiply : 

1.  "^x^  +5:^2  +  7:^-2  by  bx^  -Ix-^^. 

2.  2,x^  -  5a;  +  1  by  2x^  +  ^x^  -  bx  —  I, 

3.  {x^  +  3a;  +  2)  {^x^  -^^  ^x -\- ^){x^  -  x  -  1). 

4.  (30;'-^  +  5^;  -  3)1 
6.  (5:?;*  -  2a;2  +  1)2. 

6.  (a;«  -  5a;2  +  3)2. 

7.  l^x^  +  bx^y  +  Ixy^  -  2y^){5x^  -  7xy  +  3^2). 

8.  {3x^  -  bxy  +  2/2)  (2x4  _^  3^y  _  ^^yZ  _  y^^ 

9.  (22;2  —  xy  -\-  2^2)3. 
10.  (5a;*  -  32;2^2  _^  2y4)2. 

Detached  Coefficients  in  Division. 
358.  Model  E. 

Divisor.     3-2  +  5-3 
Quotient.  2  +  3-1 


Dividend. 

6  +  5  +  1  +  11  - 
6-4  +  10-6 

-14  +  3 

9  -  9  +  17  • 
9  -  6  +  15  - 

-14  +  3 
-9 

-3  +  2- 
-3  +  2- 

-5  +  3 
-5  +  3 

The  answer  being  2a;2  +  3^;  —  1. 


DETACHED   COEFFICIENTS, 


331 


EXERCISE    CLXXII. 

Use  detached  coefficients  in  dividing  the  following : 

1.  {ISx^  -  45a;3  +  82a;2  -  67a;  +  40)  -^  {Zx^  -  4x  +  5). 

2.  (:?;*  -  6:^3  +  9a;3  -  4)  -^  (a;2  -  3a;  +  2). 

3.  (V  +  0)3^  -  ^xY  +  19^:^^  ~  15y^)  -^  (a;^  +  Sa;?/  -  5y2). 

4.  (32a;4  _|.  5^^3  _  gi^4)  ^  (2a;  +  3^). 

6.  (V  -  a;4  +  «^  -  ^^  +  i«J  -  1)  -^  (^^  -  1). 

6.  (12~38a;+82a;2-112a;3+106a;4-70a;5)-f-(7a;2-5a;  +  3). 

7.  (9a;  +  3a;*  +  14a;3  +  2)  -^  (1  +  5a;  +  x^). 

8.  (8a;8  -  16a;6  -  34a;*  +  32a;''^  -  6)  ~  (2a;*  -  7a;2  +  3). 

9.  (a;*  +  4a;3y  +  ^x'^y'^  +  Sa;?/^  +  2?/*)  -h  (a;^  +  3a;y  +  2^^), 
10.  (5a;*+  2a;3^  -  20a:y-  230;^^-  6/)-r-(5a;2+  7a;2/  +  2y^). 


Detached  Coefficients  in  H.  C.  F. 

359.  The  method  of   detached  coefficients  often  saves 
time  in  the  process  of  H.  C.  F. 

Model  F.     H.  0.  F.  of 
2a;*  -  7a;3  +  12a;2  -  Ho;  +  4 

Q2-7  +  12-11  +  4 

©3-8+5+2-2 

(D  6  -  21  +  36  -  33  +  12  ®  X   3 

®6-  16  +  10 +  4-4  (2)X2 

*®  5  -  26  +  37  -  16  ®  -  ® 

*®  8  -  23  +  22  -  7  ®  +  ® 

©10-52  +  74-32  ®  X   2 

®  2  -  29  +  52  -  25  ®  -  ® 

®  8  -  116  +  208  -  100  ®  X   4 

©93-186  +  93  ®-® 

*([!)  1-2  +  1  (f6)H-93 

(12)  16  -  46  +  44  -  14  ®  X   2 

(13)11-  20  +  7  +  2  @-® 

©88-160  +  56  +  16  @  X   8 

*@  93-186  +  93  (14)  +  ® 


3a;*  -  Sx^  +  bx?  +  2a;  -  2. 
3-8+5  +  2-2 


1-2+1  H.C.F. 

Ans, 
\f  -  2a;  +  1] 


332 


ALGEBEA, 


Detached  Coefficients  in  Elimination. 

360.  Space  and  perplexity  may  be  saved  in  solving  equa- 
tions of  more  than  two  unknown  letters  by  using  a  method 
of  detached  coefficients;  that  is,  by  omitting  the  letters  and 
arranging  the  coefficients  of  the  same  letter  always  in  th^ 
same  column. 

Model  G.     ^x  -]-  ^  -  bz  +    w  =  2^0 
2x  -  'dy  -\-    z  -  4:tu  =  -  3 
7x  +  7y  -  3z  +  2w  =  93 


© 
® 
® 
® 


@276 
@393 


X  - 
2 
3 
7 
1 


y  +  4:z  +  3io  =  27 


-  5 
1 

-  3 
4 


30 
-    3 

93 

27 


®   6 

-  9 

3 

-  12 

-  9 

®  X  3 

®  13 

-  2 

* 

-  iO 

84 

@  +  ® 

®   8 

-  12 

4 

-  16 

-12 

®  X  4 

®   7 

-  11 

* 

-  19 

-39 

®-® 

®  10 

-  15 

5 

-  20 

-  15 

®  X  5 

@  13 

-  13 

* 

-  19 

15 

®  f  ® 

®  13 

-  2 

-  10 

84) 

First 

®   7 

-  11 

-  19 

-39  [ 

New 

®    13 

-  13 

-  19 

15  j 

Set 

(Q)   6 

-  2 

* 

54 

®-  ® 

®    26 

-  4. 

-  20 

168 

®  X  2 

@  13 

9 

-  1 

153 

@  -  ® 

@  130 

90 

-  10 

1530 

@  X  10 

®  117 

92 

* 

1446 

©-  ® 

@   6 

-  2 

54) 
1446  j 

Second 

©117 

92 

New  Set 

~  92 


2484 
3930 


(Q)  X  46 
@  +  (!D 


DETACHED   COEFFICIENTS.  333 

From  @,x=ilO 

Subst.  in  @,  60  -  2y  =  54;  2/  =  3 

Subst.  in  ®,  130  —  6  —  IQw  =  84:;  w  =  4: 

Subst.  in  ®,  10  -  3  +  43/  +  12  =  27;  y  =  2 

ic  =  10 
2^  =  3 

z  =  2 

W   =    4r 


Coefficients  of  Powers. 

361.  The  successive  powers  of  {a  +  ^)  maybe  calculated 
by  detached  coefficients.     Thus  we  obtain : 

1  +  1 
1  +  1 
1  +  1 
1  +  1 

1  +  2  +  1     (second  power) 

1  +  1 
1  +  3+1 

1  +  a  +  i 

1  +  3  +  3  +  1     (third  power) 

' 1  +  1 

1+3+3+1 

1+3+3+1 
1  +  4  +  6  +  4  +  1     (fourth  power) 

Since  the  multiplier  is  always  1  +  1,  the  set  of  coefficients 
for  the  next  higher  power  can  always  be  obtained  from 
those  of  a  given  power  by  using  them  twice  as  a  partial 
product,  displaced  as  usual  one  term  to  the  right.     Thus 


334 


ALOEBRA. 


the  calculation   for  successive   powers   would    appear  as 
follows : 


3d 

4tli 

5tli 

6tli 

7tli 

8tli 

9th 

lOth 

llth 

12tli 


1+  3+  3+    1 

1+  3+    3+    1 
1+  4+  6-h    4-f    1 

1+  4+    6+    4+    1 
1+  5+10+  10+    5+     1 

1+5+  10+  10+     5+  1 

1+  6+15+  20+  15+    6+  1 

1+  6+  15+  20+  15+  6+    1 

1+  7+21+  35+  35+  21+  7+     1 

1+  7+  21+  35+  35+  21+     7+ 


1 


1+  8+28+  56+  70+  56+  28+     8+     1 

1+  8+  28+  56+  70+  56+  28+    8+     1 
1+  9+36+  84+126+126+  84+  36+    9+     1 

1+  9+  36+  84+126+126+  84+  36+    9+  1 
1+10+45+120+210+252+210+120+  45+  10+  1 

1+10+  45+120+210+252+210+120+  45+10+  1 
1+11+55+165+330+462+462+330+165+  55+11+  1 

1+11+  55+165+330+462+462+330+165+55+11+1 
1+12+66+220+495+792+924+792+495+220+66+12+1 


THE   BINOMIAL   THEOREM. 

362.  The  coefficients  of  any  power  of  the  binomial 
a  -\-  h  are  determinable  by  a  famous  rule,  known  as  the 
Binomial  Theorem. 

In  the  first  place  these  coefficients  are  the  same  whatever 
letters  we  use  instead  of  a  and  Z>,  and,  in  the  enunciation 
of  the  rule  here  adopted,  it  will  help  us  at  first  to  get  the 
coefficients  of  powers  of  a;  +  1. 

363.  In  {x  +  1)^  the  first  term  is  cc^. 

364.  For  the  other  coefficients : 

Multiply  the  coefficient  of  any  term  by  the  index  and 
divide  by  the  number  of  the  term,  to  get  the  coefficient 
of  the  next  term. 


TSE  BINOMIAL  THEOREM.  335 

Model  H.— In  writing  (x  +  ly^: 
the  first  term  is  x^^. 

The  coefficient  of  this  term  is  1; 

the  index  is  13; 

it  is  the  first  term. 
IX  13 -7-1  =  13  (coefficient  of  the  2d  term). 
the  second  term  is  13^;^^. 

The  coefficient  of  this  term  is  13; 

the  index  is  12; 

it  is  term  number  2. 

13  X  12  -^  2  =  78  (coefficient  of  the  3d  term). 
the  third  term  is  HSx^^, 

The  coefficient  of  this  term  is  78; 

the  index  is  11; 

it  is  term  number  3. 

78  X  11  -^  3  =  286  (coefficient  of  the  4th  term). 
and  so  on. 


EXERCISE  CLXXIII. 

Write  the  first  four  terms  of 

1.  {x  +  ly,  2.  {x  +  1)17.  3.  {x  +  1)26. 

4.  {x  +  iy^  5.  (:^  +  l)^* 

6.  Write  all  the  terms  of  {x  +  iy\ 

7.  A^erify  the  coefficients  obtained  by  the  rule  for  {x  +  iy 
and  {x  +  ly. 

8.  Find  the  value  of  {x  -\-  ly  when  x=l;  also  the  value 
of  {x  + 1)\ 

9.  What  is  the  sum  of  the  coefficients  of  any  power  of 
the  binomial  x  -{-1? 

10.  How  many  terms  in  the  expansion  *  of  {x-\-iy? 

11.  The  10th  term  of  (a  +  by^  is  92378aioj9;  find  the 
coefficient  of  the  next  term. 

*  Where  an  algebraic  expression  of  one  term  can  be  written  out  as 
a  series  of  terms,  that  series  is  called  the  expansion  of  the  expres- 
sion. 


336  ALGEBRA. 

12.  The  8th  term  of  (a  +  ^)i6  is  llMOa^Z*^;  find  the 
coefficient  of  the  next  term. 

13.  The  11th  term  of  {a  +  hf'  is  184756«io^iO;  find  the 
coefficient  of  the  next  term. 

14.  The  6th  term  of  («  + Z^)^^  is  142506^25^^;  find  tlie 
coefficient  of  the  next  term. 

15.  The  4th  term  of  {a  +  h)^  is  6545a32^,3.  ^^^  the 
coefficient  of  the  next  term. 

Symmetry  of  the  Coefficients. 

365.  The  coefficients  obtained  by  this  rule  increase  from 
the  first  term  towards  the  middle,  and  decrease  in  the 
same  way  from  the  middle  to  the  end,  with  exact  sym- 
metry; so  that  if  the  set  of  coefficients  were  turned  end 
for  end,  the  entire  expression  would  be  unchanged.  If 
we  write  then  the  expansion  of  (1  +  xY  beginning  at  the 
EKD,  with  x^y  and  understand  by  ^'the  number  of  the 
term  "  the  number  counting  fkom  the  end,  the  rule  will 
give  the  same  set  of  coefficients. 

N^ow  the  expression  {a  +  hy  has  of  course  the  same 
coefficients  as  {x  +  1)"  and  (1  +  xy.  In  applying  the 
rule,  if  we  count  from  the  beginning,  we  must  understand, 
by  the  words  '^the  index,"  the  index  of  a\  and  if  from  the 
other  end,  we  must  understand  the  index  of  Z>. 

EXERCISE    CLXXIV. 

Find  the  first  three  and  the  last  three  terms  of : 
1.  {x  +  ^)^^  2.  {a  +  l)"^.  3.  {a  +  x)"^. 

4.  {x  +  aY\  5.  (^+2)1^. 

Powers  of  Differences. 

366.  In  the  expansion  of  (a  —  hy  we  must  remember 
that  — •  6  is  a  negative  factor,  and  the  terms  will  conse- 
quently be  +  or  -—  according  as  an  even  or  an  odd  num- 
ber of  factors  I  are  contained  in  it;  in  other  words, 
according  as  the  index  of  l  is  even  or  odd. 


THE  BINOMIAL  THEOREM.  SSiT 

EXERCISE    CLXXV. 

First  five  terms  of: 

1.  (a  +  hf\       2.  {a  +  ly^      3.  {a  -  iy\ 

4.    («  -  ^)2i.  5.    {a  +  Z>)23. 

JFVr^^  three  and  last  three  terms  of: 

6.   (x  +  ^)i<^.  7.    (^  -  2/)^^.  8.    (a  -  xy^ 

9.    (a;  -  Z>)80.  10.    («  +  «/)iio. 

Irregular  Coefficients. 

367.  It  is  evident  that  this  rule  will  not  apply  to  the 
numerical  factors  of  the  successive  terms  in  expressions 
like  (2a;  -  3)^ 

Model  I. 
{2x  -  3)^  =  32^5  _  24:0x^  +  720^^  -  lOSOa;^  +  810a:  -  243. 

Here  32  X  5  -^  1  ^^ 240;  240  X  4  -f-  2  ^  720;  and  so  on. 

Such  expressions  are  expanded  by  first  writing  the  same 
power  of  a  simple  binomial  as  a  pattern,  and  then  substi- 
tuting.    As  in  this  case: 

(a  -  by  =  a^  -  ba'h  +  IQaW  -  10a?b^  +  ^ah^  -  W. 

Now  in  the  required  expansion  we  must  have  2a;  instead 
of  a,  and  3  instead  of  Z>;  and  making  those  substitutions 
we  obtain  the  expansion  as  above. 

EXERCISE  CLXXVI. 


In  the  same  way 

expand : 

1.   (3a:  -  by. 

4.  (2a;  -  \y.^ 

7.  (Sa^  -  2a;)' 

2.  (2«  -  h'^y. 

6.    (3.  +  1)^ 

».(^+2-r 

»■  {-  -  ii 

(a       2\« 

9.(2+^3)'- 

10.    {p 

-  5g)5.           11.   {Zh 

-  Uf. 

338  ALGEBRA. 


The  Binomial  Theorem  Generalized. 

368.   In  generalizing  the  binomial  theorem,  we  obtain 

by  the  rule: 

n{n  -  l){n  -  2){n  -  3) 
+  1.2.3.4  ^ 

w(w  -  l)(w  -  2){n  -  3)(n  -  4) 
■•"  1.2.3.4.5 

w(n-l)(w-2)(w-3)(w-4)(w-5) 
^~  1.2.3.4.5.6  +  -  •  • 

or,  with  two  letters : 

7K^-l)(n-2)(7i~3) 
+  1.2.3.4  "^     ^ 

.  ^(->^  ~  1)(^  -  2)(.^  -  3)(^  -  4) 
'     +  1.2.3.4.5  "^     ^+... 

The  coefficient  for  a  term  further  on  in  the  expansion 
becomes  very  unwieldy,  and  a  generalized  formula,  for  any 
term  of  any  power,  would  be  very  hard  to  remember.  To 
simplify  the  expression  a  new  symbol  is  introduced. 


Factorial  n. 

369.  The  factorial  of  any  given  number  is  the  product 
of  all  the  positive  integers,  beginning  with  1  and  ending 
with  the  given  number. 

The  factorial  of  5  is  written  5!  or  |5,  and  read  ^'factorial 
five/' 

6!  =  1.2.3.4.5  =  120 
6!  =  1.2.3  .4.5.6  =  720 


THE  BINOMIAL  THEOREM,  339 

Higher  factorials,  like  20 !,  give  numbers  that  are  incon- 
veniently large  in  the  ordinary  notation.  In  figuring  the 
values  of  such  coefficients  it  is  best  to  express  them  with 
prime  factors,  in  the  order  of  magnitude.     Thus : 

20 !  =  1.2.3.2^5.2.3.7.2^3'.2.5.11.2«.3.13.2.7.3.5.2*.17.2.3M9.2«.5 
=  2i8.38.54.7'».ll. 13.17.19 

370.  In  finding  the  values  of  expressions  involving  fac- 
torials, the  work  of  calculation  can  often  be  shortened  by 
factorization. 

8! 
Model  J. — In  the  expression     ^ 

the  factors  5.4.3.2.1  are  common  to  numerator  and 
denominator;  hence 

8! 

1;  =  8  .  7  .  6  =  336 

5! 

Model  K. 
10!  -  5!  =  51(10.9.8.7.  6-  1)  =  120  X  30239 

EXERCISE    CLXX^MI. 

Find  the  values  of  the  following  expressions  : 

11!  11!  -  7!  3!(8!-  6!) 

^'  ^'  *•         6!       •  ^-  5! 

10!  ill__iiL  ?'_!: 

^-  5T5T*  ^'  3T8!       2!  9!'  ^'  5!      5!' 

20!  -  18!  100!  -  99! 

8. 


'•  187(10!  8!  5!)*  **'       3(99!)      ' 

87!      _  13!  -  11! 

^'  29(86!)  ^^'      55(9!)    * 

The  symdol  8 !  is  in  some  modern  loohs  largely  replaced 
by  18;  the  symbols  have  the  same  meaning^  and  loth  are  in 
good  use.     They  are  used  indifferently  in  the  following : 


340  ALGEBRA, 

11.  What  factors  must  be  multiplied  into   8.7.6  to 
give  18  ?     [Express  in  briefest  form.] 

12.  What  factors  must  be  multiplied  into  10  .  9  .  8  to 
give  111  ? 

13.  What  factors  must  be  multiplied  into  20  .  19  .  17  to 
give  1 20^? 

14.  What  factors  must  be  multiplied  into   \n  to  give 
|(^  +  2)? 

15.  What  factors  must  be  multiplied  into  n{n—l)(n—%) 
to  give  \n  ? 

Find  the  value  of  n  in  the  following  equations  : 

{n  —  1)1  {n  —  b)\ 

n\  ^^  105(^-4)!       ^      ,    ^ 

17.  7 ^TT-i  =  ^2.  20.   -y-^^ wrv-  =2^  +  5. 

{n  —  2)\  {n  —  d)\  ' 

(^  +  1)!       ..A  f:\7^^(n-l)      1 

18.  / ^^rff  =  110.  21.        '-^ —  - 


-4^ 


'•  (w  -  1) !  ~  '"  \|  |w  ~  3  ■ 

i«+_2    _     |w  +  l    _  9 
Uw  -  2  ■  I5|w  -  4  ~  4  ■ 


4!(w-2)!.     (w-3)!     _    48! 


24. 


(^  -  6) !   •  46 !  (^  -  5) !  47  ' 

{n  +4)(^  ~  5)!  _  {n  -  4)(y^  +  3)!  _  5 
(^  -  4) !  (^^  +  4) !        "6"* 

[(^  -  1)  H'       (^  +  1) !  (^  -  1) !  ~ 

For  Any  Term  of  Any  Power. 

371.  To  generalize  the  binomial  theorem,  so  as  to  be 
able  to  write  any  term  of  any  power  directly  from  a  for- 
mula, let  us  first  write 


THE  BmOMTAL  THBOBEM.  341 

where  «,,  a^^  a^,  a^,  etc.,  represent  the  successive  coeffi- 
cients^ as  obtained  by  the  rule. 

372.  In  each  term  there  is  a  number  (in  the  third  term, 
for  example,  it  is  2)  which  appears  in  three  places: 

it  is  the  index  of  y ; 

it  is  subtracted  from  n  to  give  the  index  of  x\ 

it  is  the  suffix  of  a. 
This  number  is  called  the  modulus  of  the  term  and  is 
represented  by  h', 

in  the  first  term  h  =  0, 

in  the  second  term  k  =  1, 

in  the  third  term  ^  =  2, 

in  the  fourth  term  h  =  Z, 

and  so  on;  in  general, — 

373.  The  modulus  of  any  term  is  one  less  than  the 
number  of  the  term. 

374.  We  may  now  state,  and  will  hereafter  prove. 

The  Binomial  Formula: 


CCi-  = 


^^^  M{n  -  k)\' 
Model  L.— The  7th  term  of  {x  +  yY^  is 

'-,<x?y^. 


6!  5! 

This  reduces  Y'^^s!'/'!  =  ^-3.7.11  =  462   as  pre- 
viously found. 


Model  M.— The  4th  term  of  (x  -  yf^  is 

3  !  17  !^  ^  • 

20    19    18 
The  coefficient  reduces  to       '  _  '       =  20 .  19 .  3  =  1140. 


342  ALGEBRA. 

Model  N.— The  13th  term  of  {a  -  If'  =  +  ^^  !  13  f''^''' 


=  2« .  5^  7  .  17 .  19  .  23. 


121  13!  ~  1.2.3.  4.5.  6.  7.8.9.10.11.12 

Ans.  22 .  5^  7  .  17 .  19 .  23an^^ 

EXERCISE    CLXXVIII. 

By  means  of  the  formula  write : 

I    (^  -  ^)22;     9th  term.            6.  {a  +  bf^;  12th  term. 

2.  (x  -  y)'^y     7th  term.            7.  {x  +  y)^-,  6th  term. 

3.  {x  —  yY^j  6th  term.  8.  {x  —  yf^]  10th  term. 
4^  (^p^qY^-^  11th  term.  9.  {h  +  ky^;  14th  term. 
5^  (^_^.)i8;     8th  term.          10.  {s  -  tf^;  13th  term. 

11.  {a  -  V2)23;  10th  term. 

/  I  \  28 

12.  [a^ 1    ;  8th  term. 

/  1   \^^ 

13.  [a  —  ~-^     ;  7th  term. 


Va^'    ' 

/-\20 

3 


14.  fa;  -  i^)    ;  12th  term. 


19 


15.  (-2  ""  ^j    >  ^^^  term. 

/   X  1   y 

16.  — = 7^     ;      13th  term. 

V  V2y        Vx  / 

17.  (-^-J^)'';      7th  term. 
\  V2x        6a  I 

18. TT-)    ;  11th  term. 

\  a        dxj 

I                 1      y^ 
10.  \2x^y —]    ;  15th  term. 

V  ^yVxl 

20.   ^  \/x-  |/^]'';      14th  term. 


TEE  BINOMIAL   THEOREM,  343 

375.  According  to  the  Binomial  Formula, 

^    ^    ^  '       ^       k\{n  —  ]c)\ 

Now  if  we  could  prove, - 

First:  that  the  formula  gives  the  correct  coeffi- 
cients for  the  first  few  powers  of  («  +  h)',  and, 

Secondly :   that  if  the  formula  gives  the  correct 

coefficients  for  any  one  power  of  {a  +  Z>)  it  must 

also  give  the  correct  coefficients  for  the  next  higher 

power; 

then  we  should  have  to  admit  that  the  formula  was  valid 

for  all  powers  oi  {a  -{-  h). 

Proof  of  the  Binomial  Formula. 

376.  Eef erring  to  the  calculation  of  successive  powers  of 
a  binomial,  earlier  in  this  chapter,  where  the  method  of 
detached  coefficients  was  used,  it  is  easy  to  see  that  in 
passing  from  one  power  of  [a  -\- 1)  to  the  next  higher  power, 
the  following  rule  holds : 

In  any  power  of  {a  +  i),  starting  with  a  term  of  given 
modulus,  if  we  add  to  its  coefficient  the  preceding  coeffi- 
cient, we  obtain  the  coefficient  of  the  term  having  the 
same  modulus  in  the  next  higher  power  of  (a  -j-  h), 

377.  In  {a  -f  5)^,  therefore,  if  the  formula  holds  good, 

n\ 


the   coefficient  of   the  term  of  modulus  h  is 
and  the  coe 
is  ^  —  1,  is 


k\(:n-ky: 
and  the  coefficient  of  the  preceding  term,  whose  modulus 


ijc  -  l)\{n  -  k-\-  1)!* 


344  ALGEBRA. 

The  coefficient  of  the  term  of  modulus  Jc,  in  (a  +  hY'^^, 
would  consequently  be 

n\  I  ^' 

{k  -  l)\{n-h+  1)!+  k\{n  -  k)\ 

This  reduces: 

_  n\  n+\       _       (^+1)! 


-  (y^- 1) l{n-k) ! '  k{n+l-k) -kl{7i  +  l-k)l 

which  is  the  coefficient  that  the  formula  would  give. 

378.  Thus  we  have  shown  that,  if  the  binomial  formula 
holds  for  {a  +  Z>)",  it  holds  also  for  {a  +  by  +  \  Now  the 
formula  does  hold  for  (a  +  Z>)^,  {a  +  by,  and  so  on,  as  far 
as  we  choose  to  test  it  among  the  low  powers  of  a  -{-  b. 
According  to  our  proof,  if  it  holds  for  {a  +  by,  it  holds 
for  {a  +  by-y  and  if  for  {a  +  by,  then  also  for  {a  +  by; 
then  also  for  {a  +  ^)^  and  so  on,  step  by  step,  through  all 
powers  of  (a  +  ^)  f^^  which  the  indices  are  positive  in- 
tegers.* 

From  the  Formula  to  the  Rule. 

379.  In  the  expansion  of  (x  +  yY  the  term  of  modulus 

71 ! 
k  is,  by  the  formula,  r—~ — '—m^"  ~  V]  and  the  next  term, 
-^  kl{n  —  k)l        ^  '  ' 

which  has  for  a  modulus  k  -{-  1,  is 


y,n  ~  k  -  l^.fc  +  1 , 

y      y 


(k  +  l)l{n-k-l)r 
^  Tliis  method  of  proof  is  called  Mathematical  Induction. 


THE  BINOMIAL  THEOREM,  345 

the  numerical  coefficients  of  these  terms,  being  represented 
by  a^  and  ^^^.^i  respectively,  are  in  the  ratio 

%4.i  n\  ^  n\ 


aj,    ~  (^  +  l)l{n  -h  -  1)!   *   'k\(n-lc)\ 
_  Ic'.in  —  lc)\  _^n  —  h 

-  (^  + 1) !  (^  -  .^  - 1) ! ""  f+t: 

Whence  we  obtain  aje  +  i  =  -,  a^c]  and  this  agrees  with 

the  form  of  the  theorem  first  enunciated,  because,  in  the 
term  of  modulus  h,  7i  —  h  is  the  index  of  Xy  and  ^  +  1  is 
the  number  of  the  term. 

Negative  and  Fractional  Indices. 

380.  Thus  far  we  have  considered  the  binomial  theorem 
only  for  cases  where  7i  was  a  whole  number,  and  a  positive 
number.  It  holds  also,  with  certain  exceptions,  for  nega- 
tive and  fractional  values  of  u]  but  in  such  cases  it  always 
gives  an  infinite  series. 

In  such  cases  also  we  mast  alter  the  binomial  formula; 
for  if  n  is  fractional,  or  negative,  n\  and  {n  —  k)\  become 
meaningless  symbols. 

381.  The  coefficient  Uj^  may  always  be  written  with  k\ 
for  its  denominator,  and  with  the  product  of  h  factors  for 
its  numerator, — these  h  factors  being 

n{n  —  l){n  —  2){n  —  3)  .  .  .  {n  -  h  +  1) 
That  is:     ,^  =  -i--^)in-2)         jn-k+l) 

382.  We  may  verify  the  formula  in  some  cases  as  fol- 
lows: 


Model  0.     Vl  —  X  =  {1  —  x)i]  expand  in  a  series. 
By  the  formula :  {1  —  xY  =  1  —  Ix  —  \x^  —  ^^x^  —  . 


346 

Otherwise : 


ALGEBRA. 


|/1_ 

-  a; 

a; 

x^ 

1 

1 

""  2  " 

8 

— 

-  ^ 

X 

- 

-  ^  + 

4 

"■  2 

x^ 

a;2 

4 

2 

—  a;  - 

8 

a-2 

a;3 

a;* 

4 

8 

64 

._3 

8   ^ 

64 

2 

—  X  - 

x^ 
4 

16 


Model  P.-Expand  (1  +  o;)"^  -  ^  ^  /^  ^  ^2' 

By  the  formula:  {l+x)-^^l-2x-\-?^x^-^x^-^bx^- 
Otherwise :        1  -\-  "Zx  -{-  x^ 


1  —  2a;  +  3a:^  —  ^x^  +  bx^  — 

i 

\J^^X  +  X^ 

—  2x  —  x^ 

—  2x  —  4:X^  —2x^ 
+  3x^  +  2a;3 

+  3a;^  +  6a;^  +  3a;^ 

-  4:X^  -  'dx^ 

—  4:X^  —  8a;^  —  4a;5 
6x^  +  4a;^ 


Model  ft. — Expand  {x  +  y) 


(_2)(_3)(-4) 


^ic-2-  2aj-3y +  3aj- V  -  4.'r- V+  •  •  • 


THE  BINOMIAL   TUEOEEM,  347 

Model  R.— Find  the  11th  term  of  {x  —  y)-^. 

(-a(-i)(-s(-a(-a(-f)(-f)(-f)(-f)(-iL.-,.( ,,,. 

1  .  2  .  3  .  4  .   5   .   6   .  7   .  8   .  9   .  10        \       U) 
_J_  4  .  7  .  10  .  13  .  16  .  19  .  22  .  25  .  28  ■  31   gj  ip 
—  310*1.2.  3.  4.  5.  6.7.  8.  9.10*^  ^^ 

_215.7.11. 13.19.31      ^^Q       _  2^5.7.11.13.19.31  ^lo  f ^2 
3^*  -^uf--    -  31^  '—^0^ 


EXERCISE    CLXXIX. 

Expand  hy  the  Mnomial  theorem,  to  four  terms: 
1-   i^+y)'^'       2.   (^+l)i       3.   (rt+Z>)-ii     4.    {x—y)-^\ 
6.    (l-c^)i  6.    (:?^+2)"'.     7.    (3:^+l)i       8.  (2^  +  3^^)"^. 

Expand  and  verify,  to  four  terms : 
11.   (^+y)"'-     12.  (1+^)^.     13.   (2-2:)-^     14.   (l+a;)i 
15.    (1  —  ^)-^.      16.    (\—x)-^, 

17.  In  the  series  obtained  by  the  binomial  theorem  for 
(1  —  xY^  substitute  x  ^-  \\    substitute  also  in  (\—x)~^  = 

and  compare  the  results;    verify  by  the  theory  of 

geometrical  progression. 

18.  In  the  same  series  substitute  x  =  2;  is  this  series  as 
clearly  equal  to  the  expression  from  which  it  arose  ?  Is  the 
series  obtained  by  division  any  more  accurate  ?  Why  does 
the  theory  of  geometrical  progression  fail  us  here  ? 

19.  Find  the  8th  term  in  {x  +  y)^. 

20.  Find  the  10th  term  in  (1  -  x)^. 


CHAPTER  XV. 


LOGARITHMS. 


383.  The  work  of  calculating  numerical  products, 
quotients,  powers,  and  roots  is  much  simplified  by  express- 
ing all  numbers  as  powers  of  some  single  number;  all  these 
operations  then  become  addition,  subtraction,  multiplication, 
and  division  of  the  indices.  In  this  chapter  it  is  shown  that 
such  a  simplification  is  practicable. 


Table  of  Powers  of  lO. 

384.  By  successive  applications  of  square  and 
the  following  fractional  powers  of 
10  can  be  calculated  to  three  places 
of  decimals:  IOtb;  iQi;  10^;  10^;  10^; 
lOi^;  10^;  10§;  lO^'s  10^;  10^;  lOi; 
10^;  lOH;  102;  lO^^i;  10^';  101;  lOxi. 

Thus  we  get  the  accompanying 
table,  in  which,  for  example,  IOtb 
(or  10-'^^)  is  shown  to  be  =  1.154; 
IQi^  (or  10«^2^)  =  6.494;  etc. 


Indices. 

tV 

.0625 

1 

.1250 

i 

.1667 

t\ 

.1875 

i 

.2500 

A 

.3125 

i 

.3333 

1 

.3750 

tV 

.4375 

i 

.5000 

t\ 

.5625 

1 

.6250 

f 

.6667 

H 

.6875 

i 

.7500 

If 

.8125 

f 

.8333 

i 

.8750 

n 

.9375 

cube  root 
Powers. 

1.154 
1.334 
1.468 
1.540 
1.778 
2.048 
2.154 
2.371 
2.738 
3.162 
3.652 
4.217 
4.641 
4.870 
5.623 
6.494 
6.813 
7.499 
8.660 


848 


LOGARITHMS.  349 

385.  This  table  can  be  extended  (by  other  means)  till 
every  number  between  1  and  10,  to  two  or  three  or  even 
more  decimal  places,  has  been  exhibited  as  a  power  of  10. 

Such  a  table  is  called  a  table  of  logarithms,  or,  more 
exactly,  a  table  of  common  logarithms;  for  other  tables 
might  be  constructed,  in  which  numbers  would  be  exhib- 
ited as  powers  of  some  root  other  than  10. 

In  every  such  system, — 

386.  The  common  root  of  all  the  numbers  is  called  the 


387.  The  indices  are  called  logarithms. 

Using  the  Tables. 

388.  In  four-place  tables,  the  first  two  figures  of  each 
number  are  given  in  the  column  marked  N;  and  the  third 
figure  is  given  at  the  head  of  one  of  the  ten  columns  of 
logarithms.  Thus  for  the  number  4.27  we  should  find  the 
first  two  figures  42  in  the  N  column,  and  on  the  same  line, 
in  the  column  marked  7,  we  should  find  6304.  [Decimal 
points  are  omitted.]  We  conclude,  then,  that  4.27  =  10-^^^*; 
or  in  other  words,  the  logarithm  of  4.27  is  .6304. 

Similarly,  if  we  had  the  logarithm  .5092,  to  find  the 
number  which  corresponds  to  it,  we  first  look  up  5092 
among  the  ten  columns  of  logarithms,  notice  that  it  is  on 
a  line  with  32  in  the  column  N  and  has  3  at  the  head  of 
its  own  column,  and  so  conclude  that  5092  is  the  logarithm 
of  3.23. 

EXERCISE   CLXXX. 

Find  : 

1.  log  3.73;  log  4.32  3.  log  2.97;  log  7.29 

2.  log  5.85;  log  3.54  4.  log  1.73;  log  6.4 

5.  log  8.9;  log  9 


350  ALGEBRA, 

Find  numbers  whose  logarithns  are : 
6.  .4771;  .8597        7.  .9814;  .7980       8.  .1987;  .4518 
9.  .5024;  .4886  lo.   .8645;  .9562 

Interpolation. 
389.  Model  A.— Find  log  2.573. 

This  number  is  ^^  of  the  way  from  2.57  to  2.58;  and  as 
we  find  from  the  table  that  the  log  2.57  is  .4099  and  log 
2.58  is  .4116,  we  assume  that  log  2.573  is  -^-^  of  the  way 
from  .4099  to  .4116. 

The  difference  between  two  successive  logarithms  in  the 
table  is  called  the  tabular  difference  and  is  represented  by 
D\  and  D  has  different  values  in  different  parts  of  the 
table. 

The  difference  between  the  required  logarithm  and  the 
WEAKEST  TABULAR  LOGAEITHM  is  represented  by  d. 

The  assumption  made   above  is  that  the   dift'erence   of 

two  numbers,  in  the  same  part  of  the  table,  is  proportional 

to  the  difference  of  their  logarithms.     This  assumption  is 

not  absolutely  correct,  but  leads  to  no  serious  error. 

2.573-2.57  _  ^ 

2.58-2T57~5 

3-^ 

d  =  (.3)i)  =  (.3)(17)  =  5;  log  2.573  =  .4104 
Model  B.— Find  log  3.736. 
-      log  3.74  =  .5729 
log  3.73  =  .5717 

n^       12  log  3.7'4    =  .5729 

d  =z  (.4)  X  12  =  4.8  d=         5 

log  3.736  =  .5724 
In  most  tables  of  logarithms  the  values  of  yVA  toA 
YoD,  etc.,  are  given  in  the  margin  opposite  the  portion  of 
the  table  for  which  they  are  correct,  thus  making  it  possi- 


LOGARITHMS.  351 

ble  to  determine  the  logarithm  mentally.     These  multiples 
of  —  are  called  Proportional  Parts. 

Model  C. — Find  the  number  whose  logarithm  is  .7470. 

log  5.58  =  .7466 

log  5.59  =  .7474 

D  =  S 

d=         4 

d       4 

Ans.  log  5.585  =  .7470. 

The  table  of  proportional  parts  can  also  be  used  in  this 
process. 

EXERCISE    CLXXXI. 

Find : 

1.  log  1.832  2.  log  2.471  3.  log  5.234 

4.  log  4.788  5.  log  7.323 

Find  7iumhers  whose  logaritlwis  are : 
6.  .2835        7.    .3048        8.  .9873       9.  .5747      lo.  .4891 

Characteristics. 

390.  Although  the  tables  give  only  the  logarithms  of 
numbers  between  1  and  10,  the  logarithms  of  numbers  be- 
low 1  and  above  10  can  readily  be  obtained  also^  remember- 
ing that  the  logarithm  of  a  number  is  its  index  considered 
as  a  power  of  10. 

Model  D.— Thus  395.2  =  100  X  3.952  =  10^  x  lO'^ses^ 
since  log  3.952  is,  by  the  table,  .5968. 

10^  X  10*^^^^  =  10^*^^^^ 

Hence  log  395.2  =  2. 5968 


352  ALGEBRA. 

Every  number  may  in  the  same  way  be  separated  into 
two  factors,  of  wliich  one  is  a  number  between  1  and  10, 
and  the  other  is  a  power  of  10,  positive  or  negative.    Thus 

385400  =  3.854  X  10^ 

.0853  =  8.53  -^  100  =  8.53  X  lO'^. 

Model  E.— Find  log  .003987 

.003987  =  3.987  -^  1000  =  3.987  X  lO'^ 

=    10-3    X    10-6006 

In  practice  this  subtraction  is  not  carried  out,  but  in- 
dicated in  a  contracted  form  by  writing  the  negative  sign 
over  the  characteristic,  thus:  log  .003987  —  3.6006. 

391.  The  integral  part  of  a  logarithm  is  called  the 
CHARACTERISTIC,  and  the  rest  of  the  logarithm,  comprising 
all  the  figures  on  the  right  of  the  decimal  point  and  con- 
stituting a  proper  decimal  fraction,  is  called  the  MANTISSA. 

392.  The  characteristic  may  be  found  by  counting  off 
the  places  from  the  first  significant  figure  of  the  number 
to  the  units  place. 

l^umbers  between  0  and  1  will  have  negative  character- 
istics ;  negative  numbers  have  no  logarithms. 

EXERCISE  CLXXXII. 

Find : 

1.  log  283.5  2.  log  30.13  3.  log  78.45 

4.  log  .07832  5.  log  11.037 

Find  numbers  whose  logarithms  are  : 

6.  5.8372  7.  1.1982  8.  2.6570 

9.  3.6670  10.  3.4825 


L00ABITEM8.  353 

Augmented  Logarithms. 

393.  The  confusion  which  may  easily  be  expected  to 
arise  with  the  use  of  negative  characteristics  is  avoided  by 
adding  10  to  every  logarithm  complicated  by  such  a 
characteristic.  Thus  log  .003987  would  appear  in  calcula- 
tion as  7.6006X;  the  X  indicating  an  excess  of  10  to  be 
subtracted  from  the  result. 

Calculation  by  LogaritFims. 

394.  Since  powers  are  multiplied  or  divided  by  adding 
or  subtracting  their  indices,  so  any  numbers  may  be  mul- 
tiplied or  divided  by  adding  or  subtracting  their  logarithms. 

395.  In  the  same  way  roots  and  powers  are  obtained  by 
dividing  or  multiplying  logarithms  by  the  required  index. 

--   ,  ,  _      n^     ^  ^  (398.7)(.0983)(9.837) 

Model  F.-    Calculate  (9s,o7)(38.9)(.OQ783)(7.38) 

log  98.07    =  1.9915 


log  398.7  =  2.6006 
log  .0983  =  8.9926  X 
log  9.837  =  0.9929 
12.5861  X 

log  38.9      =1.5899 
log  .00783  =  7.8938X 
log  7.38      =0.8681 
12.3433  X 

log  ans.  =  2.5861  -  2.3433  =  .2428 

=  log  1.749                        1.749  Ans. 

EXERCISE  CLXXXIII. 

Calculate : 

1.  (28.3)(.0587)(8.93)(1.1354). 

2.  (.0057)(.00342){.007893)(8496000). 

3.  (87.6)(83.9)(.000786)(508). 

4.  (98.7)(79.8)(9780)(.00789)(8.97). 

5.  (8.732)-\       6.    f8732. 


3S4  ALGEBRA, 

7.  (c00768)(53.42)3:  1.358. 

8.  ('V^867':35.84)(l,537). 

9.  (34020)2(.0000842). 

10    f(63.8)(47.02)(25.37). 

Cologarithms. 

396.  The  computation  of  expressions  involving  division 
is  much  simplified  by  the  use  of  cologarithms.  The 
cologarithm  (more  exactly,  the  arithmetical  complement  of 
the  logarithm)  of  a  number  is  10  minus  its  logarithm.  It 
is  computed  by  subtracting  each  digit  of  the  logarithm  from 
9  except  the  last,  and  that  from  10.  Thus  log  398.7 
=  2.6006  and  colog  398.7  =  7.3994  X. 

397.  If  of  two  logarithms  I^  and  l^  the  second  were  to  be 
subtracted  from  the  first,  the  same  result,  I,  —  l^,  could  be 
obtained  by  adding  to  l^  the  cologarithm  derived  from  /,, 

WITH  AK  EXCESS  OF  10  0^  ACCOUNT  OF  THE  COLOGAKITHM; 

that  is,  l^-  l^  =  {l^  +  10  -  ZJ  -  10. 

Thus  in  computing  the  expression  in  Model  F  we  have  performed 
two  additions  and  one  subtraction;  the  use  of  cologarithms  Avould 
reduce  this  to  one  addition,  as  follows  : 

•      log  398.7  +  log  .0983  +  log  9.837  -  log  98.07  -  log  38.9 
-  log  .00783  -  log  7.38        becomes 

log  398.7  +  log  .0983  +  log  9.837  +  10  -  log  98.07 
-f  10  -  log  38.9  +  10  -  log  .00783  +  10  -  log  7.38;  or 

log  398.7  +  log  .0983  +  log  9.837  +  colog  98.07 
+  colog  38.9  +  colog  .00783  +  colog  7.38 

398.  The  cologarithm  of  .00783  does  not  imply  A^q" 
EXCESS  OF  10,  because  10  has  already  been  added  to  the 
logarithm  on  account  of  the  negative  characteristic ;  that  is, 

10  -  (10  +  log)  =  -  log 

AVith  these  chane^es  the  calculation  of  Model  F  becomes : 


LOQAniTHMQ.  355 

Model  a.  log  398.7  =  2.6006 

log  .0983  =  8.9926  X 
log  9.837  =  0,9929 
1.9915  colog  98.07  =  8.0085  X 
1.5899  colog  38.9     =  8.4101  X 
7.8938  X  colog  .00783  =  2.1062 
0.8681  colog  7.38    =  9.1319  X 

log  ans.  =  40.2428  XXXX 
=      .2428 

Ans.   1.749. 

The  Logarithmic  Schedule. 

399.  One  of  the  most  important  things  in  all  loga- 
rithmic computation  is  ordekly  area]S"GEMEN^t;  and  the 
pupil  is  recommended  to  arrange  his  work  in  advance,  by 
writing  a  schedule  of  the  logarithms  before  opening  his 
tables. 


Model  H. 


(3.87)(3.087)(38.07) 
30.08 


The  schedule  would  be  made  and  filled  out  as  follows 
(wherever  a  cologarithm  is  called  for,  the  logarithm  from 
which  it  is  derived  is  written  on  the  left  of  the  schedule): 

log  3.87    =0.5877 
log  3.087  =  0.4896 
log  38.07  =  1.5806 
1.4783  colog  30.08  =:  8.5217 X 


log  fraction  =  11.1796  X 
=    1.1796 
log  t^f raction  =       .3932 

Ans.  2.473. 


356  ALGEBRA. 

In-radius  of  a  Triangle. 

400.   If  the  three  sides  of  a  triangle  are  represented  by 
a,  h,  and  c,  the  radius  of  the  inscribed  circle  is  determined 

by  the  formula  r  =A- ~ ~ -,  where 

a  -^  l  -\-  G 


260, 


Model  I.— Compute  r  for  the  triangle  a  =  253,  i  = 

c  =  315. 

a  =  253 

log  {s  - 

-  a)  =  2.2068 

6  =  260 

log  (s  - 

-  Z>)  =  2.1875 

c  =  316 

log  {s  - 

-  c)   =  1.9956 

2s  =  828 

2.6170  colog  s 

=  7.3830  X 

5  =  414 

log  r^ 

=  3.7729 

s  —  a  =  2Ql 

log  r 

=  1.8864 

5  -  5  =  154 

r 

=  76.98 

s  -  C=:     99 

EXERCISE    CLXXXIV. 

Compute  the    radius    of  the    inscribed  circle  for  the 
triangles : 


a 

b 

c 

1. 

273 

425 

628 

2. 

175 

527 

600 

3. 

217 

404 

495 

4. 

255 

407 

596 

5. 

277 

304 

315 

Roots  of  Proper  Fractions. 
401.   In  finding  roots  of  numbers  less  than  1,  where  10 
has  been  added  to  the  logarithm,  the  resulting  logarithm 
will  have  an  excess  of  some  fraction  of  10;  thus 


LOGARITHMS. 


367 


would  be  too  large  by  3  J,  and  the  subtraction  of  this  excess 
would  be  a  bother.  So  before  dividing  by  3  the  excess  is 
made  3  X  10,  and  the  result  turns  out  to  be  just  10  too 
large. 

Thus  log     .003987  =  27.6006  XXX 


log  l/. 003987  =    9.2002  X 


4/.003987  =      .1586 


EXERCISE    CLXXXV. 


Find  the  value  of: 

(8.732 


00658 


|31.56 
•   N  708  ^ 


f  .00937 


|/.000532 

83.7         ~Y 
^'    L38700000j  • 


6.  (7  +  colog  365)i 

7.  f  10  +  log  .000873. 


8.    V- log  .07 


4.    V  log  3.003 
log  5.38 


9.  Find  the  value  of 


logiV^ 


(log  8.53).        10. 


log  3.58 
Evaluate  the  following  formulce  : 


\ogh 

wheniV^=  15.75,  ^  =  21. 
If  X  log  l  =  log  iV,  find 

iV^when  x  =  7.832  and 

d  =  3.142. 


11.  Vs{s  -  d)(s  -  c){s  -  a)  a  =  19',b==  23.1 ;  c  =  37.7. 

p=:  .6;  P=:78Q;  n  =  60. 
F=1728;     7r  =  3.1416. 
a  =  log  3;  J  =  log  4;   c  =  log  5. 


12 
13 
14 
15 


*  \      s{s  —  a) 

3 


3F 
4;r 
a 


358  ALGEBRA, 

16.  Find  two  approximate  values  that  will  satisfy  the 
equation  3^'^^  +  '^^^  —  ^• 

17.  Insert  five  geometrical  means  between  8  and  128. 

18.  Find  the  value  of  V¥. 

19.  If  the  first  term  of  a  G.  P.  is  101,  and  the  eleventh 
term  is  800,  what  is  the  twelfth  term  ? 

30.   Solve  (1- :.)>»  =  gl^j. 

Notation  by  Powers  of  Ten. 

402.  Where  a  numerical  quantity  is  very  large  (such  as 
the  diameter  of  the  earth's  orbit  expressed  in  feet,  or  the 
distance  light  would  travel  in  a  year,  expressed  in  miles)  or 
when  it  is  very  small  (as  the  length  of  a  wave  of  light  ex- 
pressed in  any  ordinary  unit)  it  is  customary  and  very  con- 
venient to  express  the  number  in  question  as  a  multiple  of 
some  power  of  ten. 

The  distance  liglit  would  travel  in  a  year  is  an  astronomical  stand- 
ard of  measure;  thus  tlie  Dog-star  is  eiglit  light-years  away,  and  the 
pole-star  is  forty-seven. 

Model  J. — To  find  the  length  of  a  light-year  we  must 
multiply  186,300,  the  number  of  miles  light  travels  in  a 
second,  by  the  number  of  seconds  in  a  year,  3600  X  24  X 
365.25 

log  365.25  =  2.5626 
log  24  =  1.3802 

log  3600      =  3.5563 
log  186300  =  5.2702 
12.7663 

The  number  of  miles  in  a  light-year  would  therefore  be 
5.839  X  10^^  or  5839  X  10\ 


LOGARITHMS.  359 


EXERCISE   CLXXXVI. 

1.  Find    the  ratio  of    374.8  X  10"^  to  seven  one-mil- 
lionths. 

2.  Find  the  200th  power  of  2. 
/28!\ 


/2 

3.  Find  the  reciprocal  of  (- 

180° 

4.  Find  the  ratio  of  1''  to 


71 

~\ioo 


5.  Find  the  75th  term  in  the  expansion  of  j  1 


(-f)" 


6.  A  cubic  foot  of  water  weighs  62|^  pounds,  and  there 
are  2000  pounds  in  a  ton;  there  are  5280  feet  in  a  mile, 
and  3956.6  miles  in  the  radius  of  the  earth.  The  volume  of 
the  earth  is  given  by  the  formula  %7tR^,  and  it  weighs  5.6 
times  as  much  as  the  same  volume  of  water.  Find  the 
weight  of  the  earth  in  tons. 

Other  Logarithmic  Systems. 

403.  A  table  of  logarithms  can  be  constructed  with  any 
number,  commensurable  or  incommensurable,  as  a  base. 
The  system  which  has  for  a  base  the  famous  incommensur- 
able number 

e  ^  2.718281828  . . . 

called  the  Napierian  system,  is  of  great  importance  in  the 
higher  mathematics. 

404.  The  logarithm  of  a  number  to  a  base  other  than  10 
may  be  denoted  by  writing  the  base  as  a  subscript.  Thus, 
loge^;  log^  {x^y\ 

log^    32  =  5;  because  2^  =  32. 

logo  (i)  =  -  i;  because  9'^  =  |. 


360  ALGEBRA, 

405.   The  base  [b),  the  number  (iV),  and  the  logarithm 

{x)  are  connected  by  the  equation  (which  is  a  definition) 

Z>^  =  N, 

Model  K. — What  is  the  base  of  a  system  of  logarithms  in 
which  the  logarithm,  of  16  is  —  1^  ? 

b-h  =2 
bi=i 

Model  L. — If  the  logarithm  of  ^  is  2,  what  is  the  loga- 
rithm of  16  in  the  same  system  ? 

b^  =  ^  =  2-3 
Z>^  =  16  =  2^ 
b  =  2-§ 

6^   ^    2  "  2 

_  3a; 

2"  2    =2^ 

r?;  =  —  |.     Ans. 

Since  tlie  logaritlim  of  a  number  in  any  system  is  easily  calculated 
from  the  logaritlim  of  that  number  in  the  ordinary  system,  there  is 
no  need  of  extensive  tables  for  bases  other  than  10. 


EXERCISE    CLVXXXII. 

1.  If  the  logarithm  of  4  is  |,  what  is  the  base  of  the 
system  ? 

2.  If  logic  {x  -}-  5)  —  .75,  what  is  the  value  of  a;  ? 

3.  What  is  the  logarithm  of  10  in  a  system  whose  base 
is  .01  ? 

4.  When  log?,  81  =  —  f ,  what  is  the  value  of  ^  ? 


LOGAmTHMS.  361 

5.  When  loga.as^  =  —  1-5,  what  is  the  value  oi  x? 

6.  Find  log25  (.008). 

7.  When  p""  is  base,  what  is  the  log  ot  p^^"^^  ? 

8.  A\^hen  ^^"^  is  base,  what  is  the  log  of  p^  ? 

9.  When  4  is  base,  what  is  the  log  of  2  ?  of  8  ?  of  4  ? 
of  1?  of  ^V?  of  .03125?  of  0? 

10.  When  (  —  16  is  base,  what  is  the  log  of  \  —  \h  ?  of  I  ~  1   : 

of  f^)'?  of-?  of  M-'?  of  (- 

\yi        y        \yl  \y  _  ^ 

11.  If  log  7  is  3,  what  is  log  343  ?  log  Vl  ?  log  1^49  ? 

12.  If  Z>^  =  iVand  1)  ^-  10%  what  is  logio  iV^? 

406.     Model  M.— Find  log,  7. 

6^  =  7 

iclog,„  6  =  log^„  7 

^        log,„  6  -  .7782 
log  .8451  =  9.9270  X 
9.8911  X  colog  .7782  =  0.1089 
log  X  =    .0359 
X  =  1.0863 

Kote  that  in  this  example  we  use  the  logarithm  of  a 
logarithm  ;  thus  9. 9270  X  =  log  .8451  =  log  log  7  ;  and 
.0359  is  not  the  logarithm  we  were  asked  to  find,  but  the 
logarithm  of  that  logarithm. 

EXERCISE  CLXXXVIII. 

1.  Find  the  log  of  .01  in  a  system  whose  base  is  20. 

2.  What  is  the  base  of  a  system  of  logarithms  in  which 
the  logarithm  of  20  is  20  ?  _ 

3.  Find  log  144  to  the  base  2^3";  to  the  base  6. 


362  ALGEBRA. 

4.  For  the  system  whose  base  is  2,  write  successively  the 
characteristics  of  the  logarithms  for  all  the  natural  numbers 
from  1  to  10. 

5.  In  the  system  where  log  2  =  3,  what  is  log  3  ? 

6.  Given  log  2  =  a;  log  3  =  Z>;  find  log  2881^3^ 

7.  Solve  the  equation  16^^  =  25. 

57 

8.  Solve  the  equation  2^  =  —. 

9.  Solve  the  equation  8^  +  ^  =  (201^17)3. 

10.  Solve  the  equation  11"^  =  2  log  22. 

11.  Insert  six  geometrical  means  between  10  and  1000. 

12.  In  G.  P.  find  a  formula  for  n  when  a^  r,  and  I  are 
given. 

13.  Eind  the  sum  of  six  terms  in  G.  P.,  of  which  the 
first  is  3  and  the  last  701 , 

14.  The  Volunteer  Aid  Association,  desiring  to  equip  a 
hospital  ship,  starts  an  '^endless  chain ^'  hoping  to  get 
$30,000.  (This  plan  consists  in  sending  out  four  letters, 
each  numbered  ^^1/^  and  each  asking  the  receiver  to  do 
three  things:  1°,  to  make  4  copies,  each  numbered  one 
more  than  the  copy  received;  2°,  to  mail  them  to  friends; 
3°,  to  mail  10  cents  to  the  Association.  The  persons 
receiving  letters  numbered  20  are  to  make  no  copies.) 
Soon  the  number  of  replies  becomes  owerwhelmingly  large, 
and  notices  have  to  be  published  to  stop  further  contribu- 
tions. What  should  have  been  the  last  letter-number,  to 
cover  the  necessary  contribution  ?  "What  would  have  been 
the  amount  received?     [Assume  that  no  letter  fails.] 

15.  A  sum  of  money  invested  in  a  certain  business  ia 
expected  to  double  itself  in  10  years.  Excluding  fractional 
rates  of  interest,  what  is  the  lowest  rate  that  will  accomplish 
this  result,  assuming  interest  to  be  compounded  annually  ? 


TABLE  OF  LOGAUUHMS,  363 


TABLE   OF   COMMON  LOGARITHMS. 

Except  in  the  zero-colunm  only  the  last  three  figures  of 
every  logarithm  are  given. 

The  first  figure  of  each  logarithm  is  the  same  as  the  first 
figure  in  the  same  line  of  the  zero-column,  except  ivhere  an 
asterisk  ai^pears, 

*  Wliere  the  last  three  figures  of  the  logarithm  are  pre- 
ceded by  an  asterisk,  the  first  figure  is  the  same  as  the  first 
figure  in  the  kext  line  of  the  zero-column. 

The  computer  will  have  to  find  for  himself  the  value  of 
D  betiueen  successive  logarithms  j  but  in  the  space  to  the 
right  of  the  heavy  vertical  line  are  given  the  values  of  the 
fractions  .ID,  ,2D,  .SD,  ,J/-D,  and  ,5D  corresponding  to 
each  value  of  D.     This  is  the  table  of  proportional  parts. 


364 


TABLE  OF  LOGARITHMS. 


N 
10 

0 

1 

2 

3 

4 
170 

5 
212 

6 
253 

7 

8 

9 

D 

13  3  4  5 

0000 

043 

086 

128 

294 

334 

374 

43 

4  9  13  17  22 

11 

0414 

453 

492 

531 

569 

607 

645 

682 

719 

755 

4a 

4  8  13  17  21 

12 
13 

0792 
1139 

828 
173 

864 
206 

899 
239 

934 
271 

969  *004 
303  335 

*038  *072  *106 
367  399  430 

41 

14 

1461 

492 

523 

553 

584 

614 

644 

673 

703 

732 

40 

39 

38 
37 

4  8  12  16  20 
4  8  12  16  20 
4  8  11  15  19 
4  7  11  15  18 

15 

1761 

790 

818 

847 

875 

903 

931 

959 

987 

^014 

16 

2041 

068 

095 

122 

148 

175 

201 

227 

253 

279 

36 

4  7  11  14  18 

17 

2304 

330 

355 

380 

405 

430 

455 

480 

504 

5?,9 

18 

2553 

577 

601 

625 

648 

672 

695 

718 

742 

765 

So 

4  7  10  14  18 

19 

2788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

34 
33 
32 

3  7  10  14  17 
3  7  10  ia  Iti 

20 

3010 

032 

054 

075 

096 

118 

139 

160 

181 

201 

3  6  10  13  16 

21 
22 

3222 
3424 

243 
444 

263 
464 

284 
483 

304 
502 

324 
522 

345 
541 

365 
560 

385 
579 

404 
598 

3J 

3  6  9  12  16 

23 

3617 

636 

655 

674 

692 

711 

729 

747 

766 

784 

»0 

29 
28 
27 
?6 

3  6  9  12  15 
3  6  9  12  14 

3  6  8  11  14 
3  5  8  11  14 
3  5  8  10  13 

24 

3802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

25 

3979 

997  *014  *031 

*048  *065  *082| 

^099 

*116  *133 

26 
27 

4150 
4314 

166 
330 

183 
346 

200 
362 

216 
378 

232 
393 

249 
409 

265 
425 

281 
440 

298 
456 

^5 

2  5  8  10  12 

28 

4472 

487 

502 

518 

533 

548 

564 

579 

594 

609 

24 

2  5  7  10  12 

29 

4624 

639 

654 

669 

683 

698 

713 

728 

742 

757 

28 

22 
21 

2  5  7  9  12 
2  4  7  9  n 
2  4  6  8  10 

30 

4771 
4914 

786 

800 

814 

829 

843 

857 

871 
"Oil 

886 

900 

31 

928 

942 

955 

969 

983 

997 

*024  "Uv3o 

32 

5051 

065 

079 

092 

105 

119 

132 

145 

159 

172 

20 

2  4  6  8  10 
2  4  6  8  10 
2  4  5  7  9 

33 

5185 

198 

211 

224 

237 

250 

263 

276 

289 

302 

19 

18 

34 

5315 

328 

340 

353 

366 

378 

391 

403 

416 

428 

17 

2  3  5  7  8 
2  3  5  6  8 

35 

36 

5441 
5563 

453 

575 

465 
587 

478 
599 

490 
611 

502 
623 

514 
635 

527 
647 

539 
658 

551 
670 

15 

2  3  4  6  8 

37 

5682 

694 

705 

717 

729 

740 

752 

763 

775 

786 

H 

13  4  6  7 

38 

5798 

809 

821 

832 

843 

855 

866 

877 

888 

899 

13 

13  4  5  6 

39 

5911 

922 

933 

944 

955 

966 

977 

988 

999 

*010 

12 

11 

12  4  5  6 
12  3  4  6 

40 

6021 

031 

042 

053 

064 

075 

085 

096 

107 

117 

41 

6128 

138 

149 

160 

170 

180 

191 

201 

212 

222 

42 

6232 

243 

253 

263 

274 

284 

294 

304 

314 

325 

43 

6335 

345 

355 

365 

375 

385 

395 

405 

415 

425 

44 

6435 

444 

454 

464 

474 

484 

493 

503 

513 

522 

10 

12  3  4  5 

45 

6532 

542 

551 

561 

571 

580 

590 

599 

609 

618 

46 

6628 

637 

646 

656 

665 

675 

684 

693 

702 

712 

9 

12  3  4  4 

47 

6721 

730 

739 

749 

758 

767 

776 

785 

794 

803 

48 

6812 

821 

830 

839 

848 

857 

866 

875 

884 

893 

49 

6902 

911 

920 

928 

937 

946 

955 

964 

972 

981 

g 

12  2  3  4 

50 

6990 

998  *007  *016  *024  ^033  *042 

*050  *059  *067 

51 

7076 

084 

093 

101 

110 

118 

126 

135 

143 

152 

52 

7160 

168 

177 

185 

193 

202 

210 

218 

226 

235 

53 

7243 

251 

259 

267 

275 

284 

292 

300 

308 

316 

54 

7324 

332 

340 

348 

356 

364 

372 

380 

388 

396 

TABLE  OF  LOGARITHMS. 


365 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

1  3  3  4  6J 

55 

7404 

412 

419 

427 

435 

443 

451 

459 

466 

474 

8 

12  2  3  4 

56 

7482 

490 

497 

505 

513 

520 

528 

536 

543 

551 

57 

7559 

566 

574 

582 

589 

597 

604 

612 

619 

627 

58 

7634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

59 

7709 

716 

723 

731 

738 

745 

752 

760 

767 

774 

7 

112  3  4 

60 

7782 

789 

796 

803 

810 

818 

825 

832 

839 

846 

61 

7853 

860 

868 

875 

882 

889 

896 

903 

910 

917 

62 

7©24 

931 

938 

945 

952 

959 

966 

973 

980 

987 

63 

7993 

^000  *007  *014 

*021 

*028  ^035 

*041 

*048 

*055 

64 

8062 

069 

075 

082 

089 

096 

102 

109 

116 

122 

6 

112  2  3 

^ 

8129 

136 

142 

149 

156 

162 

169 

176 

182 

189 

66 

8195 

202 

209 

215 

222 

228 

235 

241 

248 

254 

67 

8261 

267 

274 

280 

287 

293 

299 

306 

312 

319 

68 

8325 

331 

338 

344 

351 

357 

363 

370 

376 

382 

69 

8388 

395 

401 

407 

414 

420 

426 

432 
494 

439 
500 

445 

506 

70 

8451 

457 

463 

470 

476 

482 

488 

71 

8513 

519 

525 

531 

537 

543 

549 

555 

561 

567 

72 

8573 

579 

585 

591 

597 

603 

609 

615 

621 

627 

73 

8633 

639 

645 

651 

657 

663 

669 

675 

681 

686 

74 

8692 

698 

704 

710 

716 

722 

727 

733 

739 

745 

75 

8751 

756 

762 

768 

774 

779 

785 

791 

797 

802 

76 

8808 

814 

820 

825 

831 

837 

842 

848 

854 

859 

77 

8865 

871 

876 

882 

887 

893 

899 

904 

910 

915 

78 

8921 

927 

932 

938 

943 

949 

954 

960 

965 

971 

79 

8976 

982 

987 

993 

998  *004  *009 

*015  *020  *025 

5 

0  12  2  2 

~m 

9031 

036 

042 

047 

053 

058 

063 

069 

074 

079 

81 

9085 

090 

096 

101 

106 

112 

117 

122 

128 

133 

82 

9138 

143 

149 

154 

159 

165 

170 

175 

180 

186 

83 

9191 

196 

201 

206 

212 

217 

222 

227 

232 

238 

84 

9243 

248 

253 

258 

263 

269 

274 

279 

284 

289 

85 

9294 

299 

304 

309 

315 

320 

325 

330 

335 

340 

86 

9345 

350 

355 

360 

365 

370 

375 

380 

385 

390 

87 

9395 

400 

405 

410 

415 

420 

425 

430 

435 

440 

88 

9445 

450 

455 

460 

465 

469 

474 

479 

484 

489 

89 

9494 

499 

504 

509 

513 

518 

523 

528 

533 

538 

4 

0  112  2 

^ 

9542 

547 

552 

557 

562 

566 

571 

576 

581 

586 

91 

9590 

595 

600 

605 

609 

614 

619 

624 

628 

633 

92 

9638 

643 

647 

652 

657 

661 

666 

671 

675 

680 

93 

9685 

689 

694 

699 

703 

708 

713 

717 

722 

727 

94 

9731 

736 

741 

745 

750 

754 

759 

763 

768 

773 

95 

9777 

782 

786 

791 

795 

800 

805 

809 

814 

818 

96 

9823 

827 

832 

836 

841 

845 

850 

854 

859 

863 

97 

9868 

872 

877 

881 

886 

890 

894 

899 

903 

908 

98 

9912 

917 

921 

926 

930 

934 

939 

943 

948 

952 

99 

9956 

961 

965 

969 

974 

978 

983 

987 

991 

996 

CHAPTER  XYI. 

SUPPLEMENTARY  PROBLEMS  FOR  PRACTICE  AND 
REVIEW. 

407.  The  problems  in  this  chapter  are  selected,  for  the 
most  part,  from  examination  papers,  set  by  colleges  or  in- 
stitutions of  a  similar  grade.  The  arrangement  of  them  is 
purposely  left  somewhat  irregular,  so  that  the  attitude  of 
the  pupil  towards  his  work  may  be  more  natural  than  when, 
in  doing  a  set  of  carefully  classified  exercises,  he  knows  be- 
forehand the  kind  of  difficulty  he  is  about  to  meet.  This 
is  especially  noticeable  in  the  ^^  concrete  problems  "  which 
close  the  collection. 

In  the  absence  of  other  directions,  where  an  algebraic  ex- 
pression is  given,  simplify  it  by  performing  the  operations 
indicated  :  where  an  equation  or  a  set  of  equations  are  given, 
solve  them  for  what  seem  to  be  the  unknown  letters. 

Factor  the  follotuing  expressions  : 

1.  x^  -%x-  84.  2.  (2^;  +  3)2-  {x  -  3)1 

3.  ^2  _  ^  _  2.       4.  324t^4Z>2  -  Wb\       5.  x'^  +  l^x  -  85. 

6.  d{x-\-yY—^{x^—y'^)'-x{x-\-y).      7.  x'^  —  2xy—xz-\-2yz. 

8.  x^  —  2xy  —  z^  -\-  y^.  9.  y^  +  ^if'  +  %• 

10.  ^y  —  a;^  —  y^  4-  1.  11.  a^  +  a^  —  «-  —  1. 

12.  \2x^  -  5x  -  2.      13.   12a;2  -  x-1,      u.    {x  +  1)*  -  1. 

15.  18x^-lSx^+4.x.    16.  2x^-3x^-U.     17.  x^  -  x  -  dO(j. 

18.  8:^;^  -  39a:  +  46.   19.   {x^  -  xf  -  8.     20.  x^^  -  a^\ 

366 


SUPPLEMENTARY  PROBLEMS.  367 

21.  Sex  —  12r^  +  '^ax  —  day.      22.  a^  +  4^^  +  4a^^^  —  c^ 

23.  2am —  b'^-\-m^-{-2bn-\-a'^—n^.      24.  3ax—bx—3ay-\-by, 

25.  ^^  —  20a;^  —  96?/^.  26.  2x^  +  5a;«/  —  12?/^. 

27.  «2  _  ^^J  _  ^  -  1.      28.  ^^^  -  b'\      29.  «'  +  rt'^''  -  2I)\ 

30.  G^'  --  l)(x  -  2){x  -  3)  +  (^  -  1)(:^  -  2)  -  :?;  -  1. 

31.  a^y^-\-a^x^  —  b^x^— h'^y^.  32.  6m^  —  38m  —  28. 

33.  a^  +  ^^  +  ^^'^  —  y^'  34.   (^^  +  ^^  —  ^'^Y  —  4:X^y^. 

35.  4^;^  —  (yyz  —  {9y^  +  z^).    36.   9^^  +  4^:2;  —  {4:X^  -\-  z^). 

37.  5a;^  —  15^;^  —  90a;^.        38.   a^-{-x^~{y^-]-z^)—2{yz—ax). 

39.  ^^  -  b^.      40.  a;2"*  +  i^"*  +  tV-     41.   9x^  -  37^:2  +  4. 

42.  5a;^  —  15^;^  —  902;^.  43.   {a-\-byx^  —  {a'^—b^)x—ab, 

44.  x^  —  x^  —  X  -{-  1.  45.   («^  —  V  —  c^y  —  4cb^c^' 

46.  ^^  +  (a  +  ^  +  <^)^  +  ^^  +  ^^• 

47.  Having  given  7n,  the  difference  of  the  squares  of  any 
two  consecutive  numbers,  find  the  numbers.  Verify  by 
substituting  figures  for  letters. 

48.  Give  three  different  forms  of  polynomials  that  are 
factorable,  and  illustrate  the  method  of  factoring  by  an 
example. 

49.  Find  and  factor  the  expression  which  added  to  the 
following  will  reduce  it  to  zero: 

(b  -  a){a  +  b-c)  +  {c-  b){b  +  c  -  a) 
and   check   the   result   by  substituting   a  =  ^;   &  =  —  f ; 

50.  Subtract  Sx^  —  7x  -{-  1  from  2x^  —  5x  —  3,  then 
subtract  the  difference  from  zero,  and  add  this  last  result 
to  2x^  -  2x^  —  4. 

51.  X  —  [x  —  2x  —  {x  —  3x)  —  {x  —  4a;)]. 

52.  (3a;  -  If  -  {2x'  -  l){3x  +  1) 

-  3x[{l  -  3xy  -  {2x'  -  1)  -  2{2a;  -  1)]. 


368  ALQEBBA. 

53.  If  a;  =  —  1|,  find  the  value  of 

(1  -x){l  -{-  ^x)  _  (1  +  ^)(1  -  3^) 
1  —  ^x  1  -\-^x 

54.  Find  fcur  factors  of  3(6^^  +  bxf-  10{<6x^  +  bx)  -  8. 

55.  2x  -  [a  -  (3Z>  -  6c)  +  [x  -  (4a  -  b  +  x  —  c)  +  3c] 

-  [2b  -  3a  +Tc]  ]. 
a{b  —  c){c  —  a){a  —  b)  —  b{c  —  b){a  —  c){b  —  a) 
^^'  {a  +  b){c-a){b-c)  ' 

57.  What  value  of  a  will  make  6x^  —  2x^-}-2ax^-}-2x  +  « 
an  exact  multiple  of  a^  —  x  -{-  1  ? 

b  —  4: 

58.  Assign  a  value  to  b  which  will  make equal  to 

0  ~~"  0 

b-\-6 
b-4' 

59.  Subtract  the  sum  of  the  squares  of  ax  +  by  and 
ay  —  bx  from  the  product  of  a^  +  y'^  and  b'^  -\-  xK 

60.  Supposing  that  a  ~\-  b  —  c  =  0,  simplify  T2~r — • 

4i2/  X   I  X 

61.  What  fraction  must  be  added  to  , —^7 — r—r-— ,   ^,0 

{x  —  iy{x-{-l)     (^-1)^ 

to  make  its  value  1  ? 

62.  S{x  +  z)  -  {6a  -  z)  -  2[x  -  {2a  +  z)  -  {a  —  3z)], 

X  1 

63.  Express  -^ ~  — as  a   single  fraction,  with 

X  —  y      X  —  o 

(3  —  a?)(3  +  xy  for  denominator. 

64.  Find  an  expression  such  that  when  divided  by 
lj^2a  +  3a^  +  4:a^  it  will  give  I  —  2a -\- 3a^  -  U^  for  a 
quotient,  and  5a;*  —  6^;^  for  a  remainder. 

a^  —  b^  .  a  —  b 

^^'     a^  -  6ab  +  6b^  "^  a'^  -+  3ab  -  10b'' ' 

66.  Find  the  value  of  x  from  the  proportion 

3^3  _  4^2  _  4a  +  5  *  6as+7a2  -  13a  -  15  "^  i^+^)'^' 


I 


SUPPLEMENTARY  PROBLEMS.  369 

67.  Reduce  to  a  single  fraction  in  its  lowest  terms: 

-^—-(^ ^—\ 

X""  —  X         \x^  —  1  x^  -\-  ll' 

68.  Find  the  square  root  of 

^x^  -  20r^  +  21x^  +  22.^'3  -  2^x^  -  ^x  +  9. 

69.  Reduce 


9^4  _  ^2  _p  iQ^  _  25       6:c^  -  2x^  +  lx^  +  x-b 

to  its  lowest  terms  as  a  single  fraction. 

70.    Find    the     following    expressions    in   their   lowest 
terms : 

2^  +  1 X  -\-l 

Vix^  -  lOx^  -  62;  +  4  "~  12x^  +  82:2  -  3:?;  -  2 
and 

2:r  +  1  ^  X  +  I 


l^x^  -  lOx^  -  6.f  +  4  *   12a;3  +  8:^^  -  3:?;  -  2' 
71.   Reduce  to  their  lowest  common  denominator 

and 


12x^  -  22;2  _  20a;  -  6  40;^  -  6a:^  -  4a;  +  6' 

and  find,  and  reduce  to   their  lowest  terms,  the  difference 
and  the  quotient  of  these  two  fractions. 

72.   If   -4—  =  «;    — ^  -  h]    -4—  =  ^;    find  the 

y  +  z  X  ■\-  z  x-^  y 

value  of  — h  ,— -— r  +  -7—, — . 


x^y  +  a;^f/^ 
^^  —  yH  -f-  :2/^^'^ 


73.  Find  the  value  of  ^--^    T    33  ^^^^  ^'  =  %• 


370  ALGEBRA. 

74.  Find  the  value  of  {x  -\-  y){x  —  y)(x^ -\-  y^)  when  the 
sum  of  the  squares  of  x  and  ^  is  3,  and  the  difference  of 
their  eighth  powers  is  21,  y  being  greater  than  x. 

75.  li  a  =  0,  h  =-  1^  c  =  ^y  d  =  —  \,  find  the  value  of 

'    ,    Z>2  _  ^2 

"+  :7 — ;7  +  - 


c  —  dd  —  a^a  —  h' 

76.  [x^"^—  x^—  x"^  —  x'^'iyx"'—  x^ -\-^). 

77.  What  value  of  x  will  make  x^-\-lx^-^^x^-\-Sx-\-'ib^b 
an  exact  multiple  of  2;^  —  7:?^  +  1  ? 

78.  Find  the  value  of  {a  -  h){a  -\-  I)  -  {a  -{-  bf  when 
3a  +  2c  =  4:5  and  3c  +  2a  =  15. 

79.  Find  two  expressions  whose  product  is 

12^;^  —  4:0x^  +  392;  —  9  and  whose  quotient  is  3x  —  1. 

OA    -El-   ^ +-U        1         ,x  +  2a.x  +  2b     ,  4:ab 

80.  Fmd  the  value  of  — —^^  -\ —z—  when  x  =  — --r. 

X  —  2a  ^   X  —  2b  a  +  b 

81.  Find  the  value  of 

^{y  +  ^)  +  y[^  -  {y  +  ^)]  -  ^[y  -  ^(^  -  ^)] 

when  X  =  3,  y  =  2f  z  =  1, 

2y_+jt^ 

82.  Substitute  1  —  x^  for  y  in  ^       .^  and  simplify. 

X^  —  V^  a    I    h 

83.  Find  the  value  of     o   ■      9  when  x  =  7  and 

x^  -{-  y^  a  —  b 

a  —  b 

X  *—  a      X  ~~'  b  CL 

84.  Find  the  value  of  — when  x  = 


b  a  ^        a  —  b 

85.  Find  the  value  of  ^^^  "  ^^  when  ^  +  ^/  =  8. 

2x  —  y  Ix  —  2y 

86.  Show  that  the  difference  of  the  cubes  of  any  two 
consecutive  numbers  is  one  more  than  three  times  their 
product. 


SUPPLEMENTAUY  PROBLEMS,  371 


x^  +  ^xy 
x{x  —  2i/) 


87.     What  values    has   the  expression  —2—[—^f.„  when 


3j/ 


^  —  2^/  ? 


«  U^ 


88.  What  value  must  be  assigned  to  -  in  order  that 

y 

2x-  -  xij  =  2xy  +  2/  ? 

89.  A  snm  of  1100  is  put  at  compound  interest  at  4  per 
cent  per  annum  for  o:^  years;  find  a  formula  for  the  amount. 

90.  Wluit  value  must  y  have  in  order  that  when  a;  =  1 
the  equation  'dx  -{-  2y  ~  1  =  2x  -\-  6y  —  IS  will  be  true  ? 

91.  What  value  must  be  given  to  x  in  order  that  when 
a  =  f  the  following  expressions  may  be  equal  ? 

92.  Find    the    sum    of    7-,    — ,    and     -7  ;     subtract 

be      ca  ao 

2f-  +  -  —  -J  from  the  result,  and  simplify  the  remainder. 

93.  Assuming  that  x^  -  60;^  +  ^x^  +  2Sx^  -  ^x^  -  54a; 
—  27  is  a  perfect  cube,  find  what  values  of  x  will  cause  it 
to  vanish. 

94.  Find  the  difference  of  the  squares  of  the  highest  and 
lowest  of  any  three  consecutive  even  numbers,  and  express 
the  result  as  a  theorem.  Is  this  theorem  true  of  odd  num- 
bers ? 

95.  If  the  quadratic  Ix^  —  2a;  +  (2  —  c)  is  a  perfect 
square,  what  is  the  value  of  c  ? 

96.  Prove  that  when  a  number  of  two  digits  is  equal  to 
four  times  the  sum  of  its  digits,  one  digit  is  double  the 
other. 

97.  Two  men,  working  separately,  can  do  a  piece  of  work 
in  X  days  and  y  days,  respectively ;  find  an  expression  for 
the  time  in  which  both  can  do  it,  working  together. 


372  ALGEBRA. 

98.  A  is  20  years  old,  and  B  is  —  2  years  older  ;  what 
is  the  age  of  B  ? 

99.  What  are  the  values  of  x  which  satisfy  the  equation 
x^  =  32;? 

100.  Divide  a\l)  -  c)  +  h^c  -  a)  +  c\a  -  h)  by  a-\-h  +  c 
and  factor  the  quotient. 

101.  Find  the  H.  0.  F.  of  Q^x^  -  2x^  +  9x^  +  9^;  -  4 
and  9x^  +  ^^^^  ~~  ^  I  what  value  of  x  will  make  both  these 
expressions  vanish  ? 

102.  Find  the  H.  0.  F.  of  3x^-6x^+2  and  2x^-5x^+3. 

103.  Find  the  L.  C.  M.  of  VdaP{x^  -  3a^x  +  2a^); 
Q6a^{x^  +  ax  -  2a^);  and  25b%x^  -  a^, 

104.  Find  the  G.  C.  M.  of  x^  -  93^  -  308  and 
x^  -  2l2;^+13l2;-231;  and  the  L.  C.  M.  of  12x^y{x^+y^)\ 
lSxy\x^  -  y^);  2\x\f(x'^  +  xhf  +  y^), 

105.  Find  the  H.  C.  D.  of  o;^  -  2x''  +  42^2-6:^  +  3 
and  x^  -  2x^  —  2x^  ^  <ox  -  3\  and  the  L.  C.  M.  of  3aV; 
hax'^'^  35fl^^;  and  Xhcu^x^, 

106.  Find  the  H.  0.  D.  of  x^  -  40^;  +  63  and 
^4  _  7^3  _|_  63^  _  31 .  .^^^  the  L.  C.  M.  of  "^id^x^a  -  x)\ 
2\ax(o?  —  x^) ;  12ax^{a  +  x), 

107.  Find  the  G.  0.  M.  of  15^*  +  10^^^  +  4:a^^  +  Qab^ 
-  W  and  6^3  +  19o?h  +  Sab^  -  6b^;  and  the  L.  C.  M. 
of  {s  -  ty{a  -  bf;  {t  +  sy{a^  -  P)';  {a  +  by{b  +  c). 

108.  Find  the  G.  C.  M.  of  2;5+lla;3-54and2;5+ll2;+12; 
and  the  L.  0.  M.  of  25{a^  +  P){a^  -  ¥)',  dOab(a^  +  U'); 
4.6b{a^  -  b^). 

109.  If  2;^  +  9^;  +  <^  is  exactly  divisible  by  a;  +  5,  what^ 
is  the  value  of  «  ?  I 

110.  L.  0.  M.  of  9x^  -  2:  -  2  and  4  +  72:  +  102:2  -  32;^. 

111.  Prove  that  the  G.  C.  D.  of  2  +  2;  -  2;^  -  22:^  and 
x^  —  x^  —  22^2  +  22;  is  the  square  root  of  a  factor  of  the 
second  expression. 

112.  Find  the  L.  C.  M.  of  x^  +  x^  and  x^"^  +  x^. 


I 


SUPPLBMENTART  PROBLEMS.  373 

113.  Supposing  that  x^  +  4^^  _  ^x^  __  i^o;  +  9  is  a 
perfect  square,  find  its  prime  factors. 

114.  Supposing  that  ^x^  -  12^^  j^  17^2  _  12^;  +  4  is  a 
perfect  square,  find  what  value  of  x  will  make  it  vanish. 

115.  Write  two  expressions  of  the  fourth  degree  which 
have  3a;2  _  22;  +  5  for  their  H.  C.  F. 

116.  Prove  that  y'^  —  4^  is  4  less  than  a  common  factor 
of  «/7  _  6^6  _^  Y^y5  _  12^4  _|_  4^3  and  if-  5y5  +  Sif-  4^3; 

and  find  their  H.  0.  F. 

117.  G.  C.  D.  and  L.  C.  M.  of  Qx^-bx^-lOx^-^^x-10 
and  'ix^  —  4:X^  —  9^  +  ^• 

118.  G.  0.  D.    and   L.  C.  M.  of  12^:2  -  ^^x  +  14  and 

l^x^  +  3:^  -  10. 

119.  G.  C.  M.  and  L.  0.  M.  of  Qx^  +  Ix^  -  5x  and 
152;^  +  31^3  +  102;2. 

120.  G.  C.  M.  of  2^3-15^+14  and  2a^-30aH56a-24. 

121.  G.  C.  F.  of  x^  +  6x^  +  Qx  and  3^^;^  +  7x^  +  3a;  +  2. 

122.  What  value  of  m  will  make  x^  —  {dm  —  l)x  -\-  2m 
exactly  divisible  by  ^  —  1  ? 

123.  G.  0.  D.  and  L.  C.  M.  of  xY  -  xY  and  x^+xy\ 

124.  G.C.M.  of  8:^;*-12a;3+22;2+3a;-land62;3-7a;2+l. 

125.  The  expression  x^  +  4:X^  —  2x^  —  12^  +  9  has  two 
identically  equal  factors;  what  values  of  x  will  make  it 
vanish  ? 

126.^  G.  C.  M.  and  L.  C.  M.  of  2a^  +  ha%  —  UV  +  l^ 
and  2^3  -  "^d^l  +  ^aV^  -  ¥. 

127.  Find  two  values  of  x  that  will  satisfy  both  of  the 
following  equations:   ^x^  —  1  =  ^x^  —  Qx  and  ^x^-\-l=^lx^, 

128.  G.  C.  D.  and  L.  C.  M.  of  2x^ -lx^-\-Sx^-10x-\-'d 
and  ^x^  -  lOx^  +  '^x'^  -  7^;  +  2. 

129.  H.  C.  F.  oix^  -  dx  +  2  and  x^ -\- x^  -  bx -\-  3. 

130.  Reduce  to  lowest  terms 

Qx^  -  2x^  -  11^3  +  5x^  -  10a; 
.     9x^  +  dx^  -  lla;^  +  dx^  -  10a;' 


374  ALGEBRA. 

131.  Given  the  three  expressions  2x^  -{-  x^  —  Sx"^  — -  x-{-  ; 
4:X^  +  12:^3  -x^  -  27a;  -  18;  ix'  +  Ax^  -  I7x^  -  9:r  +  18; 
find  the  G.  0.  D.  and  the  L.  C.  M.  of  the  first  two,  also 
of  the  whole  group  of  three. 

132.  Reduce  to  lowest  terms 

6x^  -  13x^  +  3:^■^  +  2x 
6x^  —  9x^  +  Iba^  -  27x  -  9  * 

d  -\-  h  ^  —  ^ 

133.  Subtract   from   the  sum  of    -^    and 


a  +  b 


the  product  of  —  ^ — - —   and  —  = ;  and  divide  the  re- 

^  0  +  a  0  —  a 

mainder  by  —   ^ 

134.  Express  as  a  theorem  the  general  law  observable  in 
the  following  equations : 


1  = 

:  1^ 

1  + 

3  = 

:  2« 

1  +  3  + 

5  = 

:   3^ 

1 

+  3  +  5  + 

7  = 

■e 

and  so 

1  on. 

135. 

Cr^ 

X         1 

~x~l 

-x\       (1  + 
+  xl   •"  \1  - 

X 
X 

\ 
-   1    . 
/ 

136. 

c 

5V\ 
T-—-).      137. 
a  J 

x^  - 
a;* 

-  10a;2  +  9 

-  4a;2  +  3  ■ 

138. 

y 

t-x 

x^  —  «/^ 

1 

1 

^<^  +  y^- 

y 

X 

too 

(b 

-  cf  +  (c  - 

of 

+  {a-  hf 

{a  -•  c){b  -  c)  +  {b  -  a){c  -  a)  +  {c  -  b){a-by 
140.  Multiply  {a  -  by  -  db  by  {a  +  bj  +  ab. 


SUPPLEMENTARY  PROBLEMS. 


375 


141.  Divide  {x  -  yY  +  {x^  -  yY  +  (^  +  vY  by 

{x  -  yf  +  x"^  -y^+{x  +  yf. 

aHh  -  cY  +  h\c  -  aY  +  cHa  -  hY 

142.  — ^^ — 4 — 1 r~-i — ^ • 

be  -\-  cci  -\-  ab 

1  1  a 

143.    z X   ^ X 


a  + 


a  + 


1 


d^  -  4 


144. 


«  +  2  '    «  -  2 

3(^2  +  f^  -  2)       3(<^2  _  ^  _  2) 


^2  -  a  -  2  a^  +  a  -  2         a^  -  4- 

145.   Divide  {d^-?M^-\-'dal)^-W){a^+2a^h  +  ^aI)^ ^W) 
by  «2  _[-  ^j  4_  2Z>2. 


146. 


147. 


2+    o 


2  +  i?; 


2  +  ' 


2  -3a; 


2-3 


2  +  :^; 


Z^^ 


a' 


2-3^ 

2 


2  + 


_2_+_^ 
2  -  3:r 


2-3 


24-a; 
2  -3a; 


-1  + 


Z^2 


«2^2       \rt2    _    ^2  -    -T   ^2   ^    J2/ 

148.   Substitute  ^  =  a2-Z>c;  B  =  V-ac\  G=c^-ah: 


^^^.  +  -  + 


149. 


150. 


(/^  +  q^){^  +  ^)^  +  ^(^^  -  qy){q^  -yy) 

(x^  +  8)[a;(a;  -  2)  -  4] 


[^2  _  2a;  +  4][a;(a;'^  -  8)  -  8]' 
151.  Express  as  a  theorem  the  fact  implied  in  the  follow- 
ing   equations  :    9^  —  7^  =  4  X  8  ;    5^  —  3^  =  4  X  4  ; 
^Z^  —  91^  =  4  X  92.      Prove    it,    and   then    express    the 
theorem  in  the  most  general  way. 


152.  Eeduce 


9.-^^  +  8 


a;  =  1;  also  when  x 


bx^  +  4' 


and   find    its  value   when 


376 


ALGEBRA, 


163.    V{a:^  +  U^){c'  +  d')  -  {ac  +  bdf- 

X  X  +  1 


154. 


x-\ 


155.  X-\-\ 


X 

^+1 


1 


i^  +  2  - 


x-\-\ 


ri        10  _    6   n 


3rt— 6 


:?;  + 


a—h 


x-\-'^ 


a  —  h- 


SU' 


a-^b 


157.   ^f 


-4bx'- 


6 


158.     I  I-  + 

2x 


a 


12ax-4:  \  'dbx^—9 


b' 


(i-*.^)-?„.|]. 


X 


a' 


X 


159. 


160. 


161. 


162. 


163. 


165. 


I  —  a/  '^  I  -[-  b  '^  a  -{-  d'^' 
'!_       2x^  +  IW  -  43:r  -  24 
_x       14a;^  --  'dlx^  —  31:^;  —  ( 


G- 


a^6'  +  abc  +  Z>^6" 


«2^'2  +  b'  * 

1    —   X^  (       X 


-\-  X 


x^  -\-  y'^  —  X  -\- 


1-/ 


.) 


1  +  g:^  1  +  a; 

1  +  a;"  1  +  a;' 

1  -  a;'^  "*"  1  -  a; 

«^  1 

a  _  1_      1  • 
6       5  "^a 


«  + 


a  + 


b  + 


2 

+ 

^t; 

1 

1 

j4 

2 

— 

^ 

*• 

1  * 

a; 

+ 

2 

1 



T 

X 

~ 

2 

1 

Z>(aZ>(?  -{-  a  +  c) 


8UPPLEMENTAEY  PB0BLEM8. 


377 


166.   Compute    the    value    of    the    continued   fraction 

1 


1    + 


4+- 


167. 


+ 


\:{x—a){x—h)       a{a  —  x)(a  —  h)       h(b—x){b  —  ay 


168.  Divide  — —7  —    o    ,    70     ^J     — ^  +   2        ir 

169.  Compute    the    value   of    the    continued    fraction 


12 


1  + 


^^\ 


170^ 


171. 


^  + 


1   - 


ah 


X  -  2 
cd 


¥  —  4^^      r(Z?  +  ^^^  * 


173. 


X 


y     z 


yz     zx     xy 


174. 
175 


.a  —  Z>      a-\-hr  ^       \a  —  b   ^  a 

«3  -  b 


a'  +  ¥  +  ab{a?  +  ¥)  +  a^Z>^  ^  ^'^  +  ¥  +  ab 
n^  -  ¥  "^       a^  -  //^ 


+  &, 


378  ALGEBRA. 

{a  -  by  -  ah{a  -  if  -  2a%^ 


176. 


(a  -  h){a^  -  ¥)  +  2a^Z>2 


g^  +   ^'  Cl_-_b  ^  ^4    _   ^2^2    ^   ^4 

^^■^^    a^   _    ^6   X   ^.   _j_   5  "^  ^4    _|.   ^2^2    _!_   ^4- 

1  «  —  5  1  rt  —  6 

^''®-     9V/y    _   1\   ~~  ^f2  ^  ry^      i      TTv  +  ^ 


2(g  -  1)      d^  _  7a  _|_  10  ^  2    a^  -  9^  +  18* 

"^'  t+yl  +  2  "^  <^  +  y)'    2' 

X  ,  X   —    1  X  —  ^ 

180.    . — ,    ow    A  + 


181. 


x+^){x-l)^     X  +  3)(2  -x)       (2-^)(l  -ic)' 

X 


ir 


1 


182-  .Tlr:TT,  +  . 


4- 3v/ 


«^  Va;  —  y      a;  +  2y/       «^  +  ^^  ~  2^*' 
2^(y  -  ])(?/  -  2)(t/  -  3)  +  1 


183. 2 

"    —  "U  ~r  -^ 

1\' 


2/"'  -  3.«/  +  1 

,,2  ™3 


184.    • ^0  X  ^3 'r-  7- ^ 

1  +  1.^'   ?:  +  i    i+M 

a;       o;^      y^  \x       y  I 


185. 


2:z:  +  4         ,        a;  +  5 


ic'^  +  7ii:  +  10  ~  ^2  +  4a;  -  5  "^  :?;^  +  :r  -  2* 

186.  3a;  -  4^+2:^=1;  4a;-2?/+3^=9 ;  2a;-3?/+42;=8. 

187.  x-\-y  —  ^z=S\  y-}-z—3x=—4c',  z-\-x—3y=  —  S. 

188.  4a;  +  2(3a;  -  ^  -  i)  =  1  +  3(y  -  1); 

^V(4a:  +  3^)  =  g+l. 


SUPPLEMENTARY  PROBLEMS, 


379 


189.  — ^1^;  3      -i«^-h2/     '^^        ;l3      ^       5      • 

190.  Find  the  values  of  a,  b,  and  c  in  the  following 
equations : 

'  3a  +  |Z»  =  30  -  c 

|(^  -c)  =  d{2b  -  7) 

3  3  19 


191. 


13 


x+2y+3  4.x-5y+Q'  6x~6y+4:      ^x+2y+l' 

2x  —  y  bx 


192.  ^x  —  5«/ 


_%.  ^  +  y  :=^ll 


7  4 

193.  4(:r-22/+4)=2:.+3(^-i);  5(^+|-)~2(^+2)  =  ll 

194.  8a;  —  ^  =  2^  +  2^  =  6. 


195. 


-  ~  2 


y  =.*^x =  y  --  2x  =  —  1, 


z         "  z 

196.  ?»x-y-\-2z=^-z-\-2x+2-^z-x+2y-b=ill. 

197.  x-b  =  \{y-  2);  4y  -  3  =  ^0-^+  10). 

198.  42;  -  6y  -  4  =  7^  +  2^  -  5  =  -  2^;  +  3y  +  23. 

^^^'  Zx~^  by      9  '     5:i;  "^  3^       4  * 

200.   — =r  +  — ^  =  4;     — =.  +  — ^  =  1. 

r^?;        Vy  Vx        Vy 

2(^2+  ^2) 


201. 


x^  y 


X  -  y 


^ab 


a^  -  b^' 

202.  2x  +  4.y  +  27z  =  28;     7x  -  Sy  -  15z  =  3; 

dx  -  lOy  -  33;2  =  4. 

203.  5a;-32/+2;2=41;    2x+y-z=17;    6x+4:y-2z=d6. 

o^.     o  1      .1  X.       .     ^^+^       21-3a;       2(2a;+3) 

204.  Solve  the  equation  6 j—  =  — c,       - 


# 
380  ALGEBRA, 


5a;--  3  _  3a;  -  5        1        8(rg  -  1) 
^^^-        4       ~      10      +20+         5        ' 

207.  1  +  o;^  =  5i  +  ?!-Ilii.     . 


7r 
208.  (7  +  i^)(8  _  a;)  -  1  =  y  +  17a;  - 


23  ,3/1  1  \       a;  -  1 

209.   .-..7—-^  +  ^(^— t  -  ^J  =  ^^1- 


23 

10(a; 


1        ,        1 
210.    — — B  + 


a;  +  8  "^  o;  +  4       a;  +  2  "^  a;  +  10" 
5a;  -  12i      3a;  -  2^  _  6a;  +  7       2a;  +  3 
^^^'  9  ^         5        ~       11      ~        8      • 

212.  3a;  -  5  +  VS{2x  -  3)(a;+l)-  (7a;  -  5) (a;  +  2)  =  0. 

213.  ?^  +  ^"  =  i[^+  1(^+1)]. 


18      ^      6 

a;2  +  1    ,        a;2  -  2  2a; 

214.   -o-^+  - 


a;^  —  a;  —  2       x  ~{-  V 
215.   (3a;  -  l)(4a;  +  5)  -  (2a;  +  3)  (5a;  -  2) 

=  (a;  -  2)(2a;  +  1)  +  12. 


216.  lo(a;  +  ^)  -  ^x[-  -  ^)  =  23. 


217. 


\a;       3 
a;  +  3      a;  --  1  __  2a; 


X  +  2"^a;  +  1        a;  —  1* 
___  a;  -  2  _  a;  +  23       10  + 


218.  a;  ~  11-3-:==^-^-^^ 

5a; +  2 

219.  ^^- 

7a 
220 


-(-^)=5^'-(4-'+4 


1        ,       3 2_  _ 


SUPPLEMENTARY  PROBLEMS,  381 

222.   3a;  -  4(2a;  +  3)2  +  190  =  0.    223.  x^  -  4.x -[-1  =  0, 

224.  ^^^  =  ^^.     225.  Vx^  +  lQx+^  =  3a;  -  5. 

226.  4a;-5  —  bx-^  -[-1  =  0. 

1       _       4  7  10       "" 


2c?;  -  3       5:?;  -  6  ^  8iz;  -  9       11:?;  -  12* 

228.  11^2  =  2OO2;  +  171. 

229.  Assuming  that  the  first  member  of  the  following 
equation  has  three  identically  equal  factors,  find  two  values 
for  a,  and  tell  why  you  cannot  find  the  other  four: 

a^  -  ISa^  +  114^4  -  288^3  -f-  228^2  -  72a  +  8  =  185193. 

1  Qy      I       A 

230.  Solve  ^-^—^+^-^  =  2^ 

231.  What  value  must  be  given  to  i  in  the  following 
equation  in  order  that  when  a  =  %  the  value  of  x  may 
be  4? 

26^2  -  h  _  a?  -  %h       3a^b  -  x)   _ 
b  -  X         x  +  b    ~^     25-^^      ~" 


232.  119  +  V4tx^  +  2^  +  7  =  12a;2  +  6x. 

233.  9.^?;  -  7  +  2Vx^  _  4:?;  -f  7  =  x^  +  6x. 

fx  +  2       X  -  2\       ^      . 

236.  3(l  +  -^)  =  -4-(l+-i^,). 

\         1  —  XI        2;4-7\         X  —  1/ 

237.  —^  +  -^  +  ^-=0. 
4  —  a;   '«  —  2       a; 


382  ALGEBRA. 

^2 

238.    ■^—  ?>x  =  x-\-  3Vx^  -  8:^;  +  9. 


•  •  .T  +  1 

240.  x^  -  2x  +  6Vx^  -  2a;  +  5  =  11. 

241.  (i?;2  -  52^)2  -  S{x'^  -  6x)  =  84. 

242.  4:X ^  =  14.  243.     i^X^  —  3;^^  +  1  =  X. 

X  —  I 

244.  ^x^y^  —  7  =  xy;  x  -\-  4:xy  =  9. 

245.  3:?; ^ = — ;  7x  —  3y  =  10. 

X  —  4/  o 

246.  X  -\-  y  =  20;  xy  =  51. 

247.  x^  +  y^=9;  x  +  y  =  3. 

248.  3{x^  +  xy)  =  4:0y;  x  —  y  =  2. 

249.  X  ^  y  =^  x^;  3y  —  X  =  y^, 

250.  i^;  —  ^  —  5;-4--=-. 
X       y        6 


251.  i?;^  —  2xy  -\-3y^  =  3;  x^ -{-  2xy  +  3y^  =  H. 

252.  29  =  ^^  +  ^i^;  14  =  ^^  +  ^±^;  2^  = 


.  --      -  .        ■,H  =  ^y- 

X    '  y 


^+y 


253.  x^-3y  +  ^^^,  =  (2:.  +  by-  Z3){x'  -  3y)  =  0. 

254.  3a;  -  2^  =  10;  x{2y  +  3)  =  6 


255. 


-^;  llx'  =  2xy  +  13y^ 


256.  x^  +  dxy  =  12  —  xy  =  IQy^  —  xy  —  x^, 

257.  2x^  +  3xy  =  12  —  xy  -\-  x^  —  IQy'^  —  xy, 

258.  x^  —  2xy  +  2/^  +  ^(^  +  ^)  "~  ^ 

=  a;(:c  -  1/  -  1)  +  ^(a;  -  ^  +  1)  =  0. 

259.  ^y  -Vy'^^y'^-  -^-^ —  =  4. 


SUPPLEMENTARY  PROBLEMS,  383 

x\  _  xy^  — 
^/  3^ 


1  /       ,  :r\       xy^  —  X      _ 
260.   5(a:y  +  -J=-.^  =  2 


a:       _       y       _        7 
*  y^  —  3  ~~  3  —  :?;^  ~~  ^^  —  a;^' 

262.  ^!+l'^^^!:^'.  25^2  +  9^2,^450. 

17  o 

263.  :2:2  +  2^^  -  4  =  7(2^2  _  3^^)  ^  35. 

264.  ^_|  = -4(2^-3:.)  =  48. 

265.  '^x^  -\-bxy  -'^  =  llxy  -  3^/2  +  1  =  20. 

266.  -  +  ^=|;   c^2  +  2i/2  =  54. 

267.  xy  —  \^\  yz  —  ^0]  xz  =  18. 

268.  y{y  -  2)  =  x{x  +  2);  y  +  2  =  X  —  2. 

269.  a;  +  2^  =r  8;  ^2  _|_  2^2  ^  22. 

270.  3  V(^  +y){^  +  1)  =  3^  +  ^  +  2  =  4. 

271.  a;2  -|-  y2  _  ^  _  y  —  7g.  xy  -\-  X  -{-  y  =  39. 

272.  4.^2  +  ^y  ==  6;  3a;y  +  ?/2  =  10. 

273.  2:2  4-  :^y  -f-  y2  —  52;  a:y  —  ic2  —  8^ 

274.  ^2  +  ^?/  —  2y^  =  7 ;  x^  —  9^2  :=  27. 

275.  {x  -  2yy+  3{x  —  2y)+2=0;  x^-2xy-3x+6y=l, 

276.  2:^;^  +  ^^  =  24;  xy  =  8. 

277.  {x+  l){y  -  2)  +  (:z;+l)2^2; 

{y  -  2)2  +  3(^  +  l){y  -  2)  =  4. 

278.  a;  +  ^  =  12;  x^  +  y^  =  74. 

279.  +  ^  =  a;  a;2  -  2/^  z=  ^>2^ 

280.  —    :  —=  3  :  7;  ^2  _  ^2  ^  9^ 


281.  2x^  +  Sxy  -3y^  +12  =  0;  3x  +  5y  +  1  =  0. 

3Vx  +  2Vy  _  ^ .    ^^  +  1  ^  ^^  -  64^ 

4:Vx  ^2  V~y 


282.    T-^ r-^  =  6;        ^g 


384  ALGEBRA. 

283.  91.^2  —  2x  =  45.  284.    x^  —  21.^2  —  100. 

285.  x'^  ~  x^  =  '7{x^  +  1). 

286.  x^  —  2x^  -\-  X  —  2  =  0. 

287.  {x  +  l){x  -  2){x^  ^  Qx  +  9)  =  0, 

2:r  -  5       3  9x  —  1       56 

2;  — 

x 


Form  the  quadratic  equations  whose  roots  are : 

290.  (a-i);    (5  +  f). 
[a  +  by    ^ 

291.  ^^ ' — r-l    ^  —  «. 

a  —  b 

292.  -  I;  f 


293.  One  root  of  the  equation  ^^  —  4:^;  +  c  is  2  +  Vs^; 
what  is  the  other  root,  and  what  is  the  value  of  c? 

294.  For  what  value  of  m  will  the  equation 

2x'^  +  dmx  +  2  =  0 

have  equal  roots  ? 

295.  Find  the  value  of  m  which  will  make 

x^  —  {^m  —  l)x  +  2m 
exactly  divisible  by  ^  —  1. 

^x  ~  a)      1        ^         \  a  I 

296.  It — ■. — (  = 2 7 — ■ -2 

b(x  -\-  a)      a  {x  +  af 

a  ~  c      X  —  a  db(x  —  c) 

29Y z=z  - 

'    X  —  a      a  —  c      {a  —  c)(x  —  a)' 
298.    x^  +  (2^2  +  '^ab  -  2^^2)2  ^  5(^2  ^  ^2)^2^ 


SUPPLEMENTARY  PROBLEMS.  385 


299.  x^  —  hxy  -\-  y'^  =  15a^: -f^  =  1. 

X        y        ^     a        h 

ah  X    '   y 

301.   Solve  the  equations 

separating  the  two  sets  of  answers  clearly  from  each  other. 

302. =  1 ;  "^xy  4-9  =  0. 

X       y 

^„              hx          a  ~  I    X       y       ^ 
303.    ^ i  ^  -TT-; I  =  ^• 

-  a       b 


ay 

bx 
y  —  h" 

a  ~  b 

X  -{-  a 

~      2     ' 

x-{-  y 

X  -  y  _ 

a^  -  W 

304.  ^^-'  -  -—^  =  -^-^;  x'  +  y'  =  2{a^  +  b^^y 
X  —  y       X  -\-  y  ah  ^  \«/ 

(  a{x  -y)  -  b{x  +  y)  =  a!'  -  b{2a  +  b); 
^"^-     (  {a  +  b)x  +  («  -  b)y  =  a{a  +  2b)  -  b\ 

306.  One  root  of  the  equation  2a;^  +  Sa;^  ~  3a;  —  2  =  0  is  1. 
Find  the  others. 

307.  Find  all  the  roots  of  the  equation  x^  =  (125)1 

308.  What   equation   leads   to   the   following   answers  : 
1,2,-3,5? 

112  1  11 

309. =  ^~  ? =  T'- 

y       X       oa    a  —  X      a  —  y      4:a 

310.  (a  —  b)x  =  (a  +  ^)y',  X  -{-  y  =  c. 
a;       1     1  +  a  ,1 

311.  -  +  -  • —  ^  +  -z' 

a       a        X  a^ 

312.  ctljx^  —  (a^  +  b'^)x  +  ab  =  0. 
X       y  X      y      a 

'''    a-J^^'^  b+a=y 

314.   {p+q)x-{p-q)y=^pq;   {p+q)x-{p-q)y  =pq. 

ax      ,       bx  .    r 

316.    1  -\ =  a  +  b. 

X  —  b       X  —  a 


386  ALGEBRA. 


X  +  a      X  —  b      2{a  +  b) 

316.  —  ; — y   —   . 

X  —  a      X  -}-  b  X 

ax^  -^  b      a  +  bx^  _  2(a^  +  b^) 
'    ax  -\-  b~^  a  —  bx  ~^    d^  —  W  ' 
318.  {a  -  b)y^  -  {a  +  b)y  +  22>  =  0. 

3:9.  fil^  +  A^  =:  1. 
1  —  px      1  -\-  px 


320.  {ax  —  bf  +  4:a{ax  —  b)  =  fa^. 

1   ,        a  a 

321.  -J-  -\ j -,  +  X ^ r  =  0. 

b   *  X  +  ab   ^  2x  -]-  ab 

322.  x^  —  {a  —  b)x  =  [c  —  a){c  —  b). 

323. 1 .  -  2=  0. 

X  —  a  ^  X  —  b 

bx  -^  c      ex  +  b      ib  -\-  c){x  +  2) 

324. ^--7  =  ^^ —    /;    . — ^• 

ax  -{-  c      ax  -\-  b         ax  +  b  +  c 

X        y        ^      X        y       2 

a        b  3«   '   6^      3 

5:^;      1  /I         \    ,   ^      d(  1  \      a;    .  Z» 

826. [--\-x]+d  =  -[cx _  — _L._. 

c        c\a         J  c\  adJ      c    '  c 

a               b          a  —  b 
327.   , =  . 

X  —  0      X  -[-  a  X 

328      I  ^^  +  ''^''  +  ^^  "  ^^^  "  ^""^^ 
(  (c  +  a)^  +  {0  —  a)y  =  2ac. 

329.  {b  —  c)x^  +  (^  —  «^)^  -\-  a  —  b  —  ^, 

330.  52;  +  «5y  =  a2  +  Z^2.   ^2  +  |2  =  -2+p- 

331.  —  4-  —  =  2:   by  —  ax  —  b^  —  a^, 
X    ^  y  ^ 

332.  a^x  —  a^^  =  2«j«,;  2a^x  +  2^^^  =  3^,^  —  a^. 
^{pq  —  a;(^  +  q)\       C^p  +  $')5'^^  _  qx  p'^q^ 


333. 


i^  +  s'  i^Ci^  +  s'?       p      {p  +  Qf 

834.   Z>:i:  +  2/  =  ^  +  ^y  =  i(^  +  ^)  +  !• 


SUPPLEMENTARY  PB0BLEM8. 


387 


335.  X  -\'  y  =  axy\  y  -\-  z  —  hyz\  z  +  x  =  cxz. 


336.  a  —  yz  . 


X 


;  y{b  —  zx)-=z-\-x;  z(c  —  xy)  —  y=x. 


337. 


338. 


2a{a  +  b)'-I)^x 
bx  —  2a 

2  I 


c^\x      2al 


4.a 


Ifx 
b 


ii-')  ' 


-b 


1 


1  - 


b 


X       a       X        b 


=  0. 


X  +  y      \ct{x  —  y)      x^  —  y^i 


y  :{lx  -  2y)  =  {b  -  a)  :  {2a  -  9b). 
[Substitute  a  =  6  and  Z>  =  —  2  in  the  answers.] 
6ax  —  4:b\ 


i[2b{x  +  1)]2 


341. 


342. 


343. 


345 


^bx^  -\-  bax 

{a  +  2b)x  _ 
a  —  2b 


s 


X       4cbx^  +  ^^- 


;)=o. 


a^      ___4^2 
a  —  2b       x' 


fa{l  +  2x)      b{3x  -  1)-|  _ 


1  +  3^       L^(l  +  3x)      a{2x  +  1) 

X  +  1       2__    ^  +  2 

cx~~  ax  —  bx'  *"*'**  a^  4"  ^^ 

1  /i?;  -  3  1\  2 


]  =  0. 


c 
ax 


abx       ^       d'^  —  b^ 
344.      o    ,    ,o— 1- 


(a''^+62)ic' 


i?;/     ; 


a^ic  —  2       «  Kd/^x  —  2      i?;  /       2^;  —  a^o;^* 

[Substitute  a  =  —  1  in  the  answers.] 

1  _  ^       ^ax?  -  Zb{x  -  2) 

^^^'  X  -"^        2a{x^  +  1)  +  36  • 

^2x  ■ 
347.  a^- 


x^2 


1  =  ^2-^  +  ^ 


348.  ax 


ax  -\-b 


2x  ' 


{b\l  +  x)x  -  ^2(1  -  x)'\  =  6. 


388 


ALGEBRA. 


349. 
350. 

OKI 

/ 

a   - 

\x 

2ax 

X 

-  4.1) 

hx 

—  a 
1 

2b 


X 

Ix  —  a 


a)~2  ~  \2   "  X  —  J' 


2abx 


2ax  -  b      2abx^  -  {2a^  +  b^)x  +  aV 


=1+-+- 

\  -\-  a-\-  X  ax 


352.   ax  ■ 


{2b  +  a)x  +  a  _  P{x  -  1) 


x^l 


4a 


353. 


354. 


355. 


X  -\-  a 
n^px  ' 

X  —  4:a      4:b  —  7a 


m^p[x  -\-  a)  ~ 
2b  —  X  —  2a  _ 

bx  ab  —  b'^      ax  —  bx 

2a'—  X  —  19^      a  —  2b  —  X        bb  —  x 


ax  —  2bx 


356.     1  — 


a^  —  46^         ax  +  2bx 
^(x_  _  2a^  -  ?>x 
a  \a  X 


=  0. 


357.   {a  +  1) — -    2  H V         , .     =  0. 

^  ;T  — 1      a  +  lL         X        a{x  —  l)J 

r   2x_       u{x  +  uy 

[jL 


358 


359. 


X-   U 

a  —U 

X  +  b  ^ 


_x  - 
2a 


3b      2a{a  -  3b)  _ 


=  0. 


2a 


X  —  b 


1-t'^ 


2a  -  h 


X 


-  b) 


360.   {x-{-a){x—b)- 


\x-\-a)     b\x-b)  _        3aW 
x-\-b 


a       {x—a)(x-[-b) 


361.   {x  +  3y):{2x-y)=i^^-lyi', 

x^=^{xy  +  3ay  +  lM^). 
[Substitute  a  =  2  and  b  =  —  3  in  the  answers.] 
a  2b  1  r       3  1 


362. 


i'  {b-a)x      L( 


y-\-^b      x—y^  {b—a)x      \_{a-\-b)y     d^—V^_ 
[Substitute  a  —  3  and  b  —.  —  1  in  the  answers.] 


=  0. 


8UPPLEMENTABT  PROBLEMS.  389 


363.  -i-f  =  1;  -+-  =  4. 
a      b  X      y 

364.  (3  +  h^){x^  -  a;  +  1)  =  (362  +  l)(;^2  +  c^;  +  1). 

4^2  4^2  _  J2  J2 

366. 


cc  +  2  "^  a:(:c2  —  4)       a;  -  2* 

X      y      -^     X      y 

2a;  + 1       3a;  + 1       1/1        2 


368. 


369. 


^  a  x\b       a, 

a  b{2x  +  1) 


:)• 


b{2x  -  1)       ^(r?;^  -  1) 

{2x  -  l){x  +  1)  ^  {2x'-l){x  -  1) 

x  +  3b      ,         3b  a  +  3b 

370.   o,..       .0..7.  +  " 


8d^-12ab   '   4a^  —  9^''^       (2«  +  3^>)(:?;  -  3Z>)" 
371.  ax-{-by  =  p;  cy  -\-  dz  =  q;  ex  -\-fz  =  r, 

X  +  13a  +  35      a-2b  _ 
^^^'    6a-3b-x  ~  x  +  2b  ~    ' 

a  +  1  a  +  4: 

373. 


{a  +  2)x  -  {a  +  3)       {a  +  6)x  -  {a  +  6) 

b  +  1  b  +  4: 


~  {b  +  2)x  -{b  +  S)       {b  +  6)x  -{b  +  6)* 
5'^*-  a  +  b+x      a'^  b'^x' 

(JL         V 

375.  -  +  -^-  =  2 ;  xy  -^  ay  —  bx  -\-  db. 

X         0 

376.  ax  ^by  =  0\  px  —  qy  =  m, 

377.  If  a,  Z>,  c,  and  cZ  are  in  G.  P.,  prove  that 

{ac  —  bd){ab  —  c^)  2«c      __ 

(ac  +  bd){ab  +  6?^)  "^  a^  +  bd  ~    '   ■ 


390  ALOEBBA. 


378.  If  «,  h,  c,  d  are  in  proportion,  prove  that 
d  c  -\-  d 


~  c^x  -\-  cdy  +  t?^2;* 


^2   1 

379.   If  fl^:  &=^  :  y,  prove  that  «^+5^:  a{a  —  b)— — ^ 


380.   If  a,  h,  p,  ^  are  in  proportion,  prove  that 
ma  -f-  "nh  :  ax  —  iy  =^  pm  -\-  qn  :  px  —  qy  \ 
if  they  are  in  coktii^ued  proportion,  prove  that 


then 


1.  Show  that  if      ,  ^,   ,      =  ^ — r^ =  i ttt-' 

a  _^  b  c 

X  -{-  2y  -{-  z  ~  2x  -{-  y  —  z  ~  ix  —  4:y  -\-  z' 


382.  If  c  :  d  =  X  :  y,  show  that  —  =:  ^^T — 5. 

•^  xy      x'^  +  ^ 

383.  Insert  two  geometric  means  between  a^""  and  J^^. 

384.  If  b,  c,  and  2b  —  a  are  in  G.  P.,  show  that  ab,  b'^, 
and  c^  are  in  A.  P. 

385.  Prove  that  if -^  =  ,,        ^7?  then  a,  b,  c,  and  J 

22:  —  Zy      2a  —  3£> 

are  in  proportion. 

386.  What  relation  exists  between  the  values  of  x  and  y 
in  the  equation  (12a;  +  y)  ^  (H^  +  ^)  ==  9  :  7  ? 

387.  The  first  and  second  terms  of  an  H.  P.  are  respect- 
ively 70  and  60;  find  the  fifth  and  seventh  terms. 

388.  An  H.  P.  has  the  same  first  two  terms  as  a  Gr.  P. 
whose  common  ratio  is  ^.  Find  the  ratio  of  their  third 
terms. 


SUPPLEMENTARY  PROBLEMS,  391 

389.  Insert  c  arithmetical  means  between  a  and  &. 

390.  Find  the  seventh  term  of  2;  |;  1;  .  .  . 

391.  Find  the  sixth  term  of  2;  f ;  1^;  .  .  . ;  also  the  sum 
of  the  first  six  terms. 

392.  Given  the  series  3f  +  IJ  +  y^^  +  .  .  .,  find  the 
sum  of  the  first  five  terms  and  of  the  whole  series. 

393.  Find  the  sum  of  nine  terms  of  the  series  2|^,  3f ,  5  .  .  . 

394.  The  first  and  third  terms  of  a  G.  P.  are  4  and  2i 
respectively;  find  the  sum  of  the  whole  series. 

395.  Find  the  sum  of  n  terms  of  the  series 

(l  -  V^)  +  (2  +  V^)  +  (3  +  3i/2")  +  •  .  • 

396.  Find  the  sum  of  n  terms:  7  +  1  +  ^  +  .  .  . 

397.  Find  the  tenth  term,  the  sum  of  ten  terms,  and  the 
entire  sum  of  the  series  whose  second  term  is  5  and  whose 
sixth  term  is  y^-^. 

398.  Find  the  sum  of  n  terms  of  an  A.  P.  of  which  the 
first  two  terms  are  a  and  h]  also  of  the  G.  P.  which  begins 
with  the  same  two  terms. 

399.  The  sum  of  five  numbers  in  continued  proportion 
is  242,  and  the  ratio  of  the  third  term  to  the  fourth  is  3 ; 
find  the  first. 

400.  If  a,  X,  and  y  are  in  continued  proportion,  and  if 

I  I  -A-n  XI,  X  x  +  a      y  +  a 

a,  X -\-  a,  y  -\-  a  are  m  A.  P. ,  prove  that  a  =  — - —  =  — - — . 

o  0 

401.  How  many  terms  of  the  series  18,  15,  12,  .  .  . 
amount  to  60  ? 

402.  If  there  are  five  numbers  such  that  the  first  three 
and  the  last  three  are  in  A.  P. ,  while  the  first,  third,  and 
fifth  are  in  G.  P.,  prove  that  the  product  of  the  first  and 
fourth,  added  to  the  product  of  the  second  and  fifth,  will 
be  double  the  product  of  the  second  and  fourth. 

403.  The  sum  of  the  squares  of  the  successive  whole 
numbers  beginning  with  1  and  ending  with  a  certain  number 
is  20  times  that  number;  find  the  number. 


392  ALGEBRA. 

404.  Find  the  sum  of  all  multiples  of  7  between  50  and 
450. 

405.  Find  the  sum  of  all  multiples  of  3  between  1  and 
3a  +  1. 

406.  Sum  the  G.  P.  27;   -  9;  ...  to  infinity. 

407.  Sum  the  series: 

I.   30  +  15  +  7i  +  3f  .  .  .  to  10  terms. 

II.   30  +  15  +  0  -  15  .  .  .  to  10  terms. 

III.  2.7  +  .09  +  .003  +  .0001  +  ...  to  10  terms. 

If  each  of  these  series  be  continued  to  infinity,  which 
will  have  a  finite  sum  ?   If  there  is  any  such,  find  the  sum. 

408.  Insert  2m  —  1  arithmetic  means  between  a  and  I. 

409.  Identify  the  series — =; =r ;  ■ -\  and 

|/2      1  +  |/2     4  +  3  V2 

write  its  fourth  term. 

410.  The  first  121  integers  can  be  arranged  in  five  groups, 
each  of  which  is  a  perfect  square,  thus:  (1);  (2,  3,  4); 
(5,  .  .  .  13);  (14,  ...  40);  (41,  .  .  .  121).  Prove  this, 
and  extend  the  property  to  a  sixth  group. 

411.  The  second  term  of  a  G.  P.  is  54,  and  the  fifth 
term  16;  find  the  series,  and  its  sum. 

412.  Find  the  sum  of  the  first  n  natural  numbers. 

413.  Find  three  geometrical  means  for  2  and  162. 

414.  Find  the  sum  to  infinity  of  the  G.  P.  J;  ^;  -J;  .  .  . 

415.  Find  the  sum  of  18  terms  of  the  series  f;   —  1; 
92. 

416.  A  person  saves  $270  the  first  year,  $210  the  second, 
$150  the  third,  and  so  on;  in  how  many  years  will  a  person 
who  saves  every  year  $180  have  saved  as  much  as  he  ? 

417.  Insert  three  geometrical  means  between  3f  and  18. 

418.  The  first  term  of  an  A.  P.  is  2,  and  the  difference 
between  the  third  and  seventh  terms  is  6.  Find  the  sum 
of  the  first  12  terms. 


SUPPLEMENTARY  PROBLEMS,  393 

419.  There  are  two  numbers  whose  geometric  mean  is  f 
of  their  arithmetic  mean;  and  if  the  two  numbers  be 
taken  for  the  first  two  terms  of  an  arithmetic  progression, 
the  sum  of  its  first  three  terms  is  36.     Find  the  numbers. 

420.  There  are  two  numbers  whose  arithmetic  mean 
is  33^^  greater  than  their  harmonic  mean.  Find  their 
ratio. 

421.  The  sum  of  three  terms  in  A.  P.  beginning  with  | 
is  equal  to  the  sum  of  three  terms  in  G.  P.  beginning  with 
1^,  and  the  common  difference  in  the  first  case  is  equal 
to  the  common  ratio  in  the  second.  What  are  the  two 
series  ? 

422.  Find  the  sum  of  ten  terms  of  the  G.  P.  in  which 
the  fourth  term  is  1  and  the  ninth  term  is  ^J^. 

423.  Find  the  ratio  of  an  infinite  geometrical  series  of 
which  the  first  term  is  1  and  the  sum  of  the  terms  f . 

424.  Find  the  sum  of  the  arithmetical  series  formed  by 
inserting  nine  means  between  9  and  109. 

425.  The  first  and  ninth  terms  of  an  A.  P.  are  5  and  22 ; 
find  the  sum  of  21  terms. 

426.  What  is  the  sum  of  the  first  200  odd  numbers  ? 

427.  Find  the  sum  of 

1  +  (1  +  ^)  +  (1  +  2^)  +  (1  +  3^)  +  . .  .  +  (1  +  nh) 

when  ^  =  2,  ^  =  11. 

428.  A  and  B  start  at  the  same  time  from  the  same 
point  in  the  same  direction.  A  goes  at  the  uniform  rate 
of  60  miles  per  day;  B  goes  14  miles  the  first  day,  16  miles 
the  second  day,  18  miles  the  third  day,  and  so  on.  At  the 
end  of  50  days  who  will  be  ahead,  and  how  much  ? 

429.  A  traveller  has  a  journey  of  132  miles  to  perform. 
He  goes  27  miles  the  first  day,  24  the  second,  and  so  on, 
travelling  3  miles  less  each  day  than  the  day  before.  In 
how  many  days  will  he  complete  the  journey  ? 


394:  ALGEBRA, 

430.  If  X  —  y  is  a  mean  proportional  between  y  and 
y  -\-  z  —  2Xy  show  that  a;  is  a  mean  proportional  between 
y  and  z, 

431.  Find  three  numbers  in  geometrical  progression 
such  that  their  sum  shall  be  14  and  the  sum  of  their 
squares  84. 

432.  What  is  the  geometrical  mean  between  2a;  —  3  and 

2X^  -\-  X^    —    4:X   —    3? 

433.  If  the  arithmetic  mean  of  two  numbers  is  -y-  and 

their  geometric  mean  is  |,  find  the  numbers. 
^  i_ 

434.  Eeduce  a"^  and  J**  to  equiradical  surds,  and  find 
their  sum,  difference,  product,  and  quotient. 

435.  Square  root  of  44  -  16  VY. 

436.  Square  root  of  ^a  -  1)  +  2  V2a^  -  7a  -  4:. 

437.  Find  the  value  of 

(35  •l/«^+  77  Va+  63  Vb  +  28  V^)(  VJc  -  Va  -  VV) 

when  ^  =  2,  &  =  3,  c=5;  and  simplify  the  result. 

438.  Square  root  of 

9a^V      3^^  +  ^g   ,  2Vc      2^aW  ,   2^6 

47    _J_    /y  

439.  Simplify  V-  a   when      \^J      =  Vl5. 

440.  When  x=  V%  find  the  value  of  ^^  -r  ^ 


441. 


{x-  If       {x  +  1)^* 


(l  +  |/2)(  1/6  -  f  3)* 


442.  |/4  +  c^  +  1^6  +  2:^  -  i^6  -  a;  =  0. 

443.  i  V'2a;  +  8  +  Vx  +  5  =  1 ;  verify  your  result. 

444.  (  Px^)  X  (  ^x^)  X  a;"^  ^  a;l 

445.  a;  Va;2  +  12  +  x  Vx^  +  6  =  3. 


SUPPLEMENTARY  PROBLEMS,  395 

446.  V^x~+T  =  V4:X  +  5  -  Vx  -  4. 

447.  2  I'iO  +  3  1^108  +  1^500  -  1^3^  -  2  ^1372. 

448.  Square  root  of 

449.  Simplify  Jl  :  ^|-;     also     {a^-^f  :  (-^)       • 


450 


451.  Simplify  {«^^)^  ^;  (~  «)2"(- a)2^+M  1^^; 

i/I08"+  4/75"-  4/27;   5^  +  3  .  5';  (2^ .  2¥. 
[Suppose  ^  to  be  an  integer.] 

452.  Find  the  value  of     1-  —  f  ^  when  x  =  .008. 

\jx 

453.  Vx  -\-  a  -\-  Vx  +  Vx  —  a  =  0;  explain  the  possi- 
bility of  satisfying  this  equation,  the  connecting  signs  both 
being  plus. 

4S4. ■    -T-    . 

485.  (3 - ^Q)y^_^_^-  vr+Tf+TT/- 

1^  _j_  2  \     X 

456.  11  +  ^s]—^-—  =  4vJ      ■   ^;  verify  your  results. 

457.  Square  root  of  a-'  +  2«-H2  -  i-^)  +  Z^'^H- 4(1  - 1?-^^). 

458.  Square  root  of  49a^  -  2Sa^  -  Yla^  +  6^^  +  f . 

459.  iVx^^  -  21/3"=  4/3(7^+4).      460.  ^^87  - 12^42^ 
461.  Find  the  value  of 

(x\a  +  y^h){y'^a  -  x^^h)  -  {xyy{a^  -  P)  +  x^ab 
when  X  =  2  and  y  =  3, 


396  ALGEBRA. 

463.  Given  (Vs  -  1)  :  4  -  a  =  4  +  a  :  3,  find  a. 
463.  Find  the  square  root  of 

-^  -  na^  +  QaW  -  Sab  +  12b^  +  %. 
¥  '  '    a^ 


464_   ,       .V       ._...-...-»,        .„    i/x  +  3+  \/x-3 


(^,^^^^'.      .65. 


Vx  +  ^  -   Vx  -  3 

466.  Square  root  of  x^  +  dx'^  —  4:X  +  lOa;^  —  12a;i. 

467.  (2  -  4/2"+  V3")(l  +  ^^  +  ^6"). 

468.  Given  x  —  Vd  +  xS/x^  —  3  =  3;  find  i?;. 

469.  Square  root  of  a%'^-  lOaJ-^- 10^-1^  + «-252+ 27. 

470.  ^^^..     471.  Value  of  (^y^32)-'. 
a^  —  0^ 

472.  Solve y + \=-  -=-%x-\-h. 

473.  Eeduce 


4/2^2  +  «§  -  6^2  |/3^2  _|_  5^^  _  2^2  ^6^2—  ll^?^  +  3^>2, 

and  find  its  value  when  a  =  3  and  Z>  =  —  1. 

474.  Simplify  x~^y  +  iz;^^"^  —  2;^;?/"^,  expressing  it  as  a 
single  fraction. 

ml  In 

475.    4 ' 

476.  Vx  —  4:+  Vx  —  11  -  V22:  +  9  =  0. 


477. 


Vx  —  Vy    I    Vx  -\-  Vy 

Vx  4-  VIJ     ^x  —  Vy 

2 
^2  ^  y2       x-y^ 

a;2  -  ^2  "*"  1_        1 


SUPPLEMENTARY  PBOBLEMS. 


397 


478.  Find  the  value  oi  4:y  —  x  when 

2V3"  V3~+  1 


1  + 


2- V3' 


^ 


V^ 


479.  |/5  -  22;  +  4/15  +  3:?;  ==  4/26  -  5a;. 

480.  rt'^^''  X   O?'^!)-^''  ~  (a3m^-2n  ><    |/p^). 

481.  Eationalize  the  denominator  of 


|/  -  3  +  3 


y  _  4  _  2  |/3 


.00    -E.-   ^  ^1,        1         .21/7  -21/343  +  7  V28  ,      , 
482.  Find  the  value  of r=^ to  two 


6|/63 


decimal  places. 


483. 


/-    5 


484.  4/3  +  o;  +  Va;  =  —=z. 

Vx 

485.  Express  with  a  rational  denominator 


y  +  V^^—  ^^ 


486.  State  the  value  of 

487.  Find  the  square  root  of 

X  —  2x^y~^  +  Sax^  +  y~*  —  Sax2/~^  +  16aV. 

488.  (7  -  4:V'd)x^  +  (^  -  i^3)^  =  2. 

489.  Find  the  value  of  2aVl  +  x?  when 


2\y^       \ja)' 


490.  ^y  ^  — 2a;*^  ^z  ^-\-z  ^ 

491.  4  \^~x-\-  Vx=  21;  verify  your  answers. 


398  ALGEBRA, 


492. 


a"-*        a^^  +  * 


493.  (256<i/~^) 

494.  '^^^  =  2i/^. 

495.  Value  of  — = to  three  decimal  places. 


496.  Vl  +  X  -  X^  _  2(1  +  ^  -  x^)  =  \. 

497.  Multiply  x  -  1^5"+  1  -  /-  10  -  2i/5 

by   a;  —  |/5~+  1  +-/—  10  —  2|/5! 

498.  Vi3  +  a;  +  4^13  -  i?;  =:  6. 

499.  2;^  —  ^i?;  :  l^a;  =  1^:^:  :  ^. 

500.  [^  -  \(\  -  |/^^)][a;  -  |(1  +  V~~d)l 

501.  ^  +  :c  =  i/^2ip~^^^2qr^. 

502.  ^v-^^^  +  ^-y 

503.  {  V~^^^  +  cf'b){  V^~^  -  c\^l). 

504.  Eationalize  the  denominator  of  -7^:r- 


Vx  -{-  Vx  -]-  y 
506.    (2;  --  5  +  2V"^^)(:^  -  5  -  2i/'^n:)» 
a       -  y^ 


506. 


1  —  X     y       -\-  X 


507.  (ai  -  a%'^  +  aiZ>§  ~  aZ>  +  aW^  -  b^){ah  +  Z>^). 

x~^  4-  aix^  4-  a^)~* 

508.    /  7=--. 

Va?  -\-  x^  -\-  aVx 


^10  ■v'aA' 


/lOl 
610.   I  — i-=  I  :  X 

V  3  4/S  V 


SUPPLEMENTARY  PROBLEMS.  399 

Find  X  and  sim- 


4 


5a  fa^    9j- 


J4  f  a^^»9     1^5 
plify  the  answer. 

511.  Divide  63;*^+^^— 19ir'^+2''+20ii;'"  +  ^— 7i^;^— 4a;^ 

by   3a;2^  -  5:^;^  +  4. 

512.  (2x'  -  1)*  -  (3^  +  1)*  =  (x-  4)*. 

513.  vl       \^i 


514.    ^134  +  84^2. 


516.  V'?  -  i?:  +    1/32;  +  10  +    -/iC  +  3  =  0. 

517.  ^^  —  a*  =  (i?;  —  Z>)^. 

518.  Which  is  the  greater,  VTo  or  f  46,  and  why  ? 

519.  Square  root  of  75  —  12  t^2l. 

520.  Solve  {x  —  af  =  {x  -  ^a){x^  +  4^2)1, 

521.  Square  root  of  41  +  12  V^. 


522.    Vx  -\-  ^  =  Vx  +  |. 


523.  Keduce  ^  +  ^  ^ ^  to  the  form  A  +  BV  -  1. 

1  +  |/-  1 


624.  V2x  -  y  =  Vx-y  +  1;  x^ -\-  4:y^  =  17. 

625.  (- 1  +  v^^y  +  (_  1  _  v'^^y. 

526.    (  V3  +  V~^^){Vd-  V~^^). 


527.  Find  the  reciprocal 


oth^ 


+  2  +  Vx"-2 


)■ 


528.  Multiply  2ya  -  V  -x  by  dV^T^  +  2^x. 

529.  Square  root  of  35  -  12  Vol 


400  ALOBBRA, 

530.    ,  + ,—  =  12. 

531.  Kationalize  the  denominators  of 

ac       ^         1  7  +  2t^6 

da^^h-^^'    a^  +  h^'    9-31/6' 

532.  Simplify 

633.    /ll  +44/6:  534.    ^5  -  ^^"247 

l^a  +  ^  |/«  —  a; 

636, 


\x       s  la       el; 


^* 


■>^- 


law       8  Ito       4  I  a;3«/5  1  Va« 

539.  Find  the  value  of  ^^  +  ^^  -      _^^- 

a^»  -  Z>3        ^^  _  ^• 

640.    \/2  +  VW. 

541.  Given  a;  =  3  -  1^11,  ^  =  3  +  j/ll ;  find  the  values 
of  the  expressions  xy; . 

542.  Vx  +  V^a  +  x  =  2  f/^  +  X. 

543.  (2  |/"=^  +  3  V~^^){4.  V"^^  -  5  1^"^=^). 

544.  Find  one  value  of  ^17       12  1^2^ 

545.  Develop  by  binomial  formula 


8UPPLEMENTABT  PROBLEMS,  401 

546.  Expand  {a^  -  U)\ 

547.  Convert      ,  into   an  infinite  series  by  the 

binomial  theorem. 

548.  Write  down  the  eighth  term  of  {a  —  ly^, 

549.  Expand  to  four  terms  Va  —  x^, 

550.  Expand  to  five  terms  (1  +  a)"^. 

551.  Expand  -7- -—r  into  a  series. 

(2a  —  3)i 

552.  Expand  to  four  terms  {a  +  x)-^, 

553.  Expand  by  the  binomial  theorem  Zb{2x  —  y)K 

554.  Write  out  the  first  five  terms  and  the  last  five  terms 
of  {x  —  yY^, 

555.  Calculate  the  sixth  term  of 


556.  Find  the  fourth  term  of 


/     Va     _       V2     Y 

VvTi/d^        3a* 'nj  ' 

hVa  _  6|/^Y' 
\   3     ~     a  }  ' 

aVa  _\" 


/' 


557.  Find  the  sixth  term  of  (  ^ 

558.  Find  the  last  four  terms  of  (^^  -  2^^)20. 

559.  Find  the  terms  which  do  not  contain  radicals  in 

/     ^  ih^' 

the  development  of  1 


(^-4)' 


560.  Find  the  prime  factors  of  the  coefficient  of  the 
sixth  term  of  the  nineteenth  power  of  {a  —  l).  What  are 
the  exponents  in  the  same  term,  and  what  is  the  sign  ? 

561.  Find  the  sixth  term  of  {x  —  i/Y;  also  of 


/   6a^ ^Y 

\7b  VI        'WaJ 


402  ALomBA. 

562.  Find  the  sixth  term  of  the  nineteenth  power  of 


i^-i} 


563.  Find  the  tenth  term  of  {x  —  yf^;  also  of 


\  vr     Va) 


564.  Write  out  {x  -  yfK 

565.  Write  out  the  first  five  terms  and  the  last  five  terms 
of  {x  —  yY^',   then  find  and  simplify  the  fifth  term  of 


566.  Find  the  sixth  and  twenty-fifth  terms  of  the  twenty- 
ninth  power  of  {x  —  y);  also  the  sixth  term  of  the  twenty- 


ninth  power  of 


\  b        2a  J' 


of 


567.  Find  the  term  which  contains  i^j*  in  (1  —  ^x)^. 

568.  Find  the  middle  term  of  {a  -  2xy^. 

569.  Find  the  middle  term  of  (1  +  .t)^". 

570.  Write  down  the  coefficient  of  x^  in  the  expansion 
1 

?/(!  +  xY 

571.  Expand  (1  —  x^)~^  to  six  terms  by  the  binomial 
theorem. 

572.  Find  the  sixth  term  and  the  ^th  term  in  the  ex- 
pansion of  {a  —  xy^". 

573.  Find  the  ratio  of  the  ^th  term  of  7- r-   to  the 

(1  —  xy 


BUPPLEMENTAnr  PROBLEMS,  403 

\/[,  y  as  expanded  by  the  binomial 

theorem. 

574.  Find  the  eleventh  term  in  the  expansion  of 

575.  Find  the  fifth  term  of  {oT^  +  2ic"^)"^ 

576.  Expand  (1  —  y)'^  to  five  terms,  and  write  down  the 
(r  +  5)th  term  in  its  simplest  form. 

577.  If  A  is  the  sum  of  the  odd  terms,  and  B  of  the 
even  terms,  in  the  expansion  of  (x  +  «)^  show  that 

578.  Write  down   the   first   four   terms,   the  last  four 
terms,  and  the  middle  term  of  {x  —  2yy^. 

579.  Expand  by  the  binomial  theorem  Ix  -\ J 

580.  Expand  (m~^  +  2n^y. 

581.  Find  the  fifth  term  of  {x-'^  -  2y^yK 

582.  Give  the  first,  third,  and  fifth  terms  in  the  expansion 


of  fx  Vy  +  :r^V' 


2Vx/ 

583.  Find  the  first  term  with  a  negative  coefficient  in 
11 
the  expansion  of  (1  +  «)   . 


584.  Fifth  term  of 

585.  Fourth  term 


fit  _  i^V 

1^  n        3'bJ  ' 

°   \{/^i       Say  ' 

/^ 3  VfV 

\2Vf         «'   /' 


586.  Fourth  term  of 

587.  Find  the  fourth  term  of  (ps^  —  3)^^ 


404  ALGBBBA. 

688.  Find  the  fourth  term  of  the  eleventh  power  of 


\3  ^     •     bVh) 

\  'da'         2 ' 


589.  Find  the  eighth  term  of  ( IZz.  —  _aVa.  h~ 

Z 

590.  Write  out  (x  —  yY^  ;  then  find,  in  its  reduced 
form,  the  fourth  term  of  this  expansion,  when 

X  —  — -=     and     v  =  -— — . 

|/^  ^  Za 

591.  Find  the  eleventh  term  in  the  expansion  of 
(28  -j-  2^^)^. 

592.  Given  log  1.3287  ^  .1234269  and  log  1.3288  = 
.1234596;  find  log  .00132874. 

593.  Given  log  3.8795  =  .5887758  and  log  38796  = 
4.5887870;  find  log  (  |/."0387957)(  1^387.954). 

594.  Find  the  value  of  7  log,  if  +  5  log.,  ff  +  3  log,  %\. 

595.  Find  the  number  whose  logarithm  is  —  2.4211522, 
having  given  log  379.18  =  2.5788454  and  log  379.19  = 
2.5788569. 

596.  Find  the  logarithm  of  6561  to  the  base  3  i^3. 

597.  Given  log  3000  =  3.4771213,  solve  the  equation 
r.Ol)^  =  .0009. 

598.  Given  log  2  ::=  .30103  and  log  3  =  .47712;  find 
the  values  of  log  1^376  and  log,  1^376. 

599.  Given  log  2.5  =  .39794  and  log  2.25  =  .35218; 
find  log  2.7  to  four  places. 

600.  Given  log  a  =  2,  log  5  =  3;  find  log^  R 


8VPPLBMENTART  PBOBLBMS.  405 


Practice  in  Forming  Equations. 

408.  In  forming  the  equation  for  any  problem  where  the 
selection  of  a  meaning  for  x  seems  difficult,  it  is  a  good  plan 
to  write  down  a  list  of  the  different  numbers  which  would 
be  mentioned  in  an  explanation  of  the  problem,  but  whose 
numerical  value  is  not  given  in  the  statement  of  the  prob- 
lem. Then  let  x  stand  for  one  of  these,  and  see  if  abbrevi- 
ations for  the  others  can  be  readily  derived ;  perhaps  it  will 
be  necessary  to  have  more  than  one  letter. 

It  may  be  that  two  different  expressions  may  be  found 
which  represent  the  same  number;  in  that  case  those  two 
expressions  would  form  an  equation. 

When  there  are  several  facts  given  in  the  statement  of 
the  problem,  it  will  be  found  that  some  of  the  facts  are 
used  in  forming  the  different  abbreviations;  when  they  are 
not  all  so  used,  the  remaining  facts,  expressed  in  algebraic 
form  by  means  of  the  abbreviations,  will  form  the  equa- 
tions needed. 

Model  A. — A  baby  gets  9  steps  away  from  her  mother  be- 
fore the  mother  starts  after  it ;  the  baby  takes  7  steps  while 
the  mother  takes  5,  but  1  of  the  mother^s  steps  is  equal  to 
2  of  the  baby^s.  How  many  steps  must  the  mother  take  to 
catch  the  baby  ? 

Here,  as  in  many  problems,  it  pays  to  draw  a  simple  dia- 
gram. 


A  B  G 

Let  ^  represent  the  mother's  position,  B  the  place  the  baby 
gets  to  before  the  mother  starts,  and  C  the  place  where  the 
mother  catches  it.  The  numbers  that  we  need  to  speak  of 
in  explaining  the  problem  would  be 

the  number  of  baby's  steps  from  B  to  0;  and 
the  number  of  mother's  steps  from  A  to  (7. 


406  ALGEBRA, 

If  we  let  X  represent  either  of  these,  an  abbreviation  for 
the  other  may  be  derived  from  the  fact  that  ' '  the  baby- 
takes  7  steps  while  the  mother  takes  5  " ;  but  for  the  answer 
asked  for,  it  would  be  more  convenient  to  let  x  =  the  num- 

^  X 
ber  of  mother's  steps  from  A  to  G,     Then  —  =:   the  num- 

D 

ber  of  baby's  steps  from  B  to  C.  Now  we  have  the  dis- 
tance from  ^  to  ^  given  equal  to  9  steps,  and  that  from 

Ix 
B  to  Coequal  to  —  steps;  but  the  distance  from  A   to  C, 

0 

which  is  equal  to  x  steps,  is  expressed  in  a  different  unit, 
namely,  mother's  steps;  using  the  third  fact  of  the  prob- 
lem, we  can  reduce  the  x  mother-steps  to  2x  baby-steps,  and 
we  get  the  equation 

2^  =  5+9. 
5 

Model  B. — A  and  B  run  a  mile.  First  A  gives  B  a  start 
of  44  yards,  and  beats  him  by  51  seconds;  at  the  second 
heat  A  gives  B  a  start  of  1  minute  15  seconds,  and  is  beaten 
by  88  yards.  Find  the  time  in  which  A  and  B  can  run  a 
mile  separately. 

Drawing  diagrams  for  the  separate  heats, 

.     44      .  1716  yds.  .       .  1672  yds.  .     88    . 


X  B  Y     Z  AW 

we  see  that  in  the  first  heat  B  ran  only  1716  yards;  in  the 
second  heat  all  the  facts  stated  refer  only  to  J.'s  running 
1672  yards.  The  numbers  that  we  need  to  speak  of  in  ex- 
plaining the  problem  are 

the  number  of  seconds  A  takes  to  run  a  mile  ; 
the  number  of  seconds  B  takes  to  run  a  mile  ; 
the  number  of  seconds  A  takes  to  run  1672  yards  ; 
the  number  of  seconds  B  takes  to  run  1716  yards. 


SUPPLEMENTARY  PROBLEMS.  407 

If  we  let  X  and  y  stand  for  the  first  two  of  these,  we  have : 

X  =  the  number  of  seconds  A  takes  to  run  a  mile  ; 
y  =  tlie  number  of  seconds  B  takes  to  run  a  mile  ; 

1673a5 

=  the  number  of  seconds  A  takes  to  run  1672  yards  ; 

"T^z—  =  the  number  of  seconds  B  takes  to  run  1716  yards. 
17d0 

Eeducing  these  fractions  to  lowest  terms  we  have 


601.  The  perimeter  of  a  right  triangle  is  11  times  as  long 
as  the  shortest  side.  What  is  the  ratio  of  the  two  sides 
containing  the  right  angle  ? 

602.  Divide  33  into  three  parts  so  that  the  first  may  be 
to  the  second  in  the  ratio  1.5,  while  the  second  is  to  the 
third  as  \  is  to  \, 

603.  The  area  of  an  oblong  floor  is  175  square  feet,  and 
its  perimeter  is  53  feet.  Writedown  and  solve  the  quadratic 
equation  which  gives  the  dimensions  of  the  floor. 

604.  A  vote  was  taken  in  a  debating  society,  and  the  mo- 
tion was  carried,  5  to  3;  on  reconsideration,  50  of  the 
affirmative  votes  deserted  to  the  negative;  if  60  more  had 
deserted  the  motion  would  have  been  lost,  3  to  4.  What 
was  the  vote  on  reconsideration  ? 

605.  A  business  man  starts  to  walk  to  his  house,  a  mile 
away;  at  the  same  moment  his  son  starts  from  the  house  to 
run  to  the  office  and  back.  The  son  being  able  to  run  in  4 
minutes  a  distance  that  the  father  walks  in  9  minutes 
where  will  he  meet  his  father,  and  where  will  he  overtake 
him? 

606.  The  number  of  months  in  a  man^s  age  on  his  birth- 
day in  the  year  1891  is  exactly  \  of  the  number  denoting 
the  year  in  which  he  was  born.    In  what  year  was  he  born  ? 


408  ALGEBRA, 

607.  A  man  who  can  row  a  miles  an  hour  in  still  water 
rows  down  a  stream  which  flows  h  miles  an  hour  and  back 
to  his  starting-place.  Show  whether  the  time  he  takes  is 
longer  or  shorter  than  the  time  required  to  row  the  same 
distance  in  still  water. 

608.  An  agent  for  a  type- writing  machine  wishes  to  ad- 
vertise that  he  will  give  one  machine  to  any  one  selling  a 
certain  number.  What  must  that  number  be  in  order  that 
the  agent  may  make  an  average  profit  of  $5,  when  the 
machines  cost  the  agent  $20  and  are  retailed  for  130  ? 

609.  A  train  goes  from  Oxford  to  London,  63  miles,  14 
minutes  faster  than  a  train  which  travels  3  miles  an  hour 
slower.     Speed  of  each  train  ? 

610.  How  much  rice  at  8  cents  a  pound  must  be  mixed 
with  20  pounds  at  11  cents  in  order  that  the  mixture  may 
be  worth  10  cents  a  pound  ? 

611.  A  fast  express  train  makes  its  entire  run  at  an  aver- 
age speed  of  51 J  miles  an  hour;  the  up-grades  at  an  aver- 
age speed  of  50  miles  an  hour,  and  the  down-grades  at  an 
average  speed  of  54  miles  an  hour ;  there  are  70  miles  more 
up-grade  than  down-grade;  what  is  the  length  of  the  run  ? 

612.  A  man  has  20  coins^  of  v/hich  some  are  half-dollars 
and  the  rest  are  nickels;  if  he  should  change  the  halves  for 
dimes,  and  the  nickels  for  cents,  he  would  have  100  coins. 
How  much  money  has  he  ? 

613.  A  miner  bought  20  dogs,  some  at  San  Francisco 
and  some  at  Juneau;  they  cost  him  on  an  average  $10 
apiece.  If  he  had  been  able  to  get  them  all  at  San  Fran- 
cisco, he  would  have  saved  $160;  if  he  had  had  to  get  them 
all  at  Juneau,  he  would  have  lost  $160.  How  many  were 
bought  at  each  place  ? 

614.  A  certain  number  exceeds  twice  the  product  of  its 
two  digits  by  73,  and  3  times  the  sum  of  its  digits  by  61; 
find  the  number. 


SUPPLEMENTARY  PROBLEMS,  409 

615.  Two  trains  start  from  opposite  ends  of  a  two-track 
road,  and  pass  each  other  6  hours  later;  one  completes  the 
run  in  5  hours'  less  time  than  the  other.  Find  the  running 
time  for  each  train. 

616.  Two  pedestrians  start  together  on  a  certain  course, 
and  one  walks  twice  as  fast  as  the  other,  over  the  whole 
course  and  back  again;  but  the  slower  one  walks  only  |  of 
the  course  and  back  again,  being  passed  by  the  faster  man 
I  mile  from  the  winning-post.  Find  the  length  of  the 
course. 

617.  The  difference  in  the  expense  of  fencing  two  square 
fields  is  ^  of  the  difference  in  value  of  the  fields  themselves; 
and  the  total  expense  of  fencing  both  fields  is  twice  the 
difference  in  value  of  the  fields.  Supposing  the  cost  of 
fencing  per  yard  happens  to  be  equal  to  the  value  of  the 
land  per  square  yard  in  either  field,  find  the  area  of  the 
fields. 

618.  A  man  buys  570  pulleys,  some  at  16  for  a  dollar 
and  the  rest  at  18  for  a  dollar;  he  sells  them  all  at  15  for  a 
dollar  and  gains  $3.     How  many  of  each  sort  does  he  buy  ? 

619.  The  difference  of  two  numbers  multiplied  by  their 
product  is  30,  and  the  difference  of  their  cubes  is  98.  Find 
the  numbers. 

620.  My  income  is  a  third  less  than  my  brother^s,  but 
my  expenses  are  40  per  cent,  less  than  his ;  and  we  each 
save  $100.     Income  of  each  ? 

621.  A  dozen  pair  of  boots  and  a  dozen  pair  of  shoes  can 
be  had  for  $46;  and  a  dozen  pair  more  of  shoes  can  be  had 
for  $105  than  of  boots  for  $100.  Price  of  each  per 
pair? 

622.  Find  three  numbers  which  are  to  each  other  as 
2:4:5,  and  such  that  the  sum  of  the  greatest  and  least  ex- 
ceeds the  other  by  21. 

623.  The  sum  of  the  squares  of  the  two  numbers  formed 


410  ALOEBBA. 

by  the  same  pair  of  consecutive  digits  is  585.     Find  the 
numbers. 

624.  My  brother  and  I  can  do  a  piece  of  work  in  6  days ; 
you  and  I  would  take  9  days  to  do  it.  In  what  time  could 
you  and  he  do  it,  supposing  that  he  works  twice  as  fast 
as  I? 

625.  A  sailor  on  shore  leave  spent  ^  of  his  wages  in 
presents  for  his  friends,  \  for  carriage-hire,  \  for  circus- 
tickets,  had  \  stolen,  and  retained  $1.50  when  he  returned 
to  the  ship.     How  much  had  he  at  first  ? 

626.  A  large  sum  of  money  passes  through  the  hands  of 
an  importing  agent,  a  custom-house  official,  and  a  lawyer, 
each  of  whom  takes  from  it  a  different  percentage  of  the 
value  it  has  when  he  receives  it;  if  the  sum  remaining  is 
$1200,  and  the  sum  taken  by  the  last  of  the  three  men  is 
$300,  $400,  or  $600,  according  to  the  different  orders  in 
which  the  money  can  be  successively  submitted  to  them, 
what  was  the  original  sum  ? 

627.  A  and  B  run  a  mile  race.  In  the  first  heat  B  re- 
ceives 4  seconds  start,  and  is  beaten  by  32  yards;  in  the 
second  heat  B  receives  64  yards  start,  and  wins  by  2  sec- 
onds.    Find  the  time  each  takes  to  run  a  mile. 

628.  A  and  B  worked  together  for  8  days,  and  completed 
half  of  a  definite  task  that  had  been  put  before  them ;  B 
then  left  the  job,  and  A,  after  working  with  C,  a  new 
workman,  for  6  days,  left  the  job  also;  then  C  worked  on 
alone,  and  completed  the  task  in  2  more  days.  What  C 
accomplished  in  the  last  two  days  was  less  than  what  A 
would  have  done  by  the  exact  amount  that  B  would  have 
done  in  one  day.  Find  the  time  in  which  each  workman 
could  do  the  work  alone,  and  the  proportions  in  which 
they  should  be  paid. 

629.  An  egg-dealer  bought  a  certain  number  of  eggs  at 
32  cents  per  score,  and  5  times  that  number  at  $1.50  per 


SUPPLEMENTARY  PROBLEMS.  411 

hundred.     He  sold  them  all  at  20  cents  per  dozen,  gaining 
$648  by  the  transaction.     How  many  eggs  did  he  buy  ? 

630.  A  number  of  soldiers  drawn  up  in  a  hollow  square 
4  ranks  deep  can  also  be  formed  in  a  solid  column  of  which 
the  number  of  ranks  will  be  equal  to  the  number  of  men  in 
each  rank ;  the  number  of  men  in  the  front  rank  being  25 
greater  in  one  formation  than  in  the  other.  Find  the  num- 
ber of  men. 

631.  A  had  :J^  as  much  money  as  B  had;  and  since  then 
he  has  paid  B  16.  The  money  A  now  has  bears  to  the  sum 
he  originally  had  the  same  ratio  that  the  money  B  now  has 
bears  to  115.  How  much  had  each  at  first  ?  Explain  the 
negative  answer. 

632.  A  man  walks  away  from  home  a  miles  per  hour, 
rests  h  hours,  and  then  walks  back  at  the  rate  of  c  miles  per 
hour,  being  gone  from  home  h  hours  in  all.  How  far 
away  did  he  get  ? 

633.  On  what  day  of  a  year  which  is  not  a  leap-year  are 
the  ^^  days  past"  and  the  ^*days  to  come"  of  a  calendar 
consecutive  squares  ?  Note  that  the  *'days  past  "  for  any 
given  date  include  that  day. 

634.  A  letter-carrier  is  delayed  half  the  time  during  the 
first  45  minutes  of  his  trip  of  6  miles;  then  he  finds  that  ho 
must  go  2  miles  per  hour  faster  to  get  through  on  time. 
How  many  miles  per  hour  is  he  expected  to  go? 

635.  At  what  time  between  1  and  2  o'clock  is  the  minute- 
hand  as  far  from  the  hour-hand  as  the  latter  is  from  12  ? 
Generalize  this  problem;  discuss  the  solution  for  different 
hours,  and  the  possibility  of  realizing  the  solutions  on  an 
actual  clock. 

636.  A  clock  is  set  right  at  noon,  January  1;  when  the 
correct  time  is  1:15  p.m.  the  same  day,  the  hands  of  this 
clock  are  6  minutes  apart.     When  will  it  be  an  hour  slow  ? 

637.  A  certain  number  divided  by  another  gives  a  quotient 


412  ALGEBRA. 

3  and  a  remainder  2;  if  9  times  the  second  number  be 
divided  by  the  first,  the  quotient  is  2  and  the  remainder  11. 
Find  the  numbers. 

638.  A  number  consists  of  two  figures  whose  product  is 
21;  if  22  be  subtracted  from  the  number,  and  the  sum  of 
the  squares  of  its  figures  added  to  the  remainder,  the  order 
of  the  figures  will  be  inverted.     What  is  the  number  ? 

639.  A  fraction  becomes  f  by  the  addition  of  3  to  the 
numerator  and  1  to  the  denominator.  If  1  be  subtracted 
from  the  numerator  and  3  from  the  denominator,  the  frac- 
tion becomes  ^.     Find  the  fraction. 

640.  Divide  111  into  three  parts,  such  that  the  products 
of  the  several  pairs  may  be  in  the  ratios  4:5:6. 

641.  A  and  B  run  a  race  of  480  feet.  The  first  heat  A 
gives  B  a  start  of  48  feet,  and  beats  him  by  6  seconds;  the 
second  heat  A  gives  B  a  start  of  48  yards,  and  is  beaten 
by  2  seconds.     How  many  feet  can  each  run  in  a  second  ? 

642.  A  and  B  set  out  at  the  same  time  to  walk  to  a  place 
6  miles  distant  and  back  again.  After  walking  for  2  hours, 
A  meets  B  coming  back.  Supposing  B  to  walk  twice  as 
fast  as  A,  find  their  respective  rates  of  walking. 

643.  A  man  bought  a  certain  number  of  eggs  for  $2.  If 
he  had  paid  5  cents  more  per  dozen,  he  would  have  received 
2  dozen  less  for  the  same  money.  How  many  dozen  did  he 
buy,  and  what  did  he  pay  for  them  ? 

644.  A  boy  spent  his  money  in  oranges.  If  he  had 
bought  5  more,  each  orange  would  have  cost  him  |-  cent 
less;  if  3  less,  |  cent  more.  How  much  did  he  spend, 
and  how  many  did  he  buy  ? 

645.  A  number  is  made  up  of  three  figures  whose  sum  is 
17.  The  figure  of  the  hundreds  is  double  that  of  the  units. 
When  396  is  subtracted,  the  order  of  the  figures  is  reversed. 
What  is  the  number  ? 

646.  The  smaller  of  two  numbers  divided  by  the  larger  is 


SUPPLEMENTARY  PROBLEMS,  413 

.21,  with  a  remainder  .04162;  the  greater  divided  by  the 
smaller  is  4,  with  .742  for  a  remainder.  What  are  the 
numbers  ? 

647.  A  number  is  made  up  of  three  figures  whose  sum  is 
17.  The  figure  of  the  units  is  f  that  of  the  hundreds. 
When  297  is  subtracted,  the  order  of  the  figures  is  inverted. 
What  is  the  number  ? 

648.  A  and  B  together  can  do  |-  of  a  piece  of  work  in  6 
days.  If  B  can  do  -J-  of  it  in  18  days,  how  long  will  it  take 
A  to  do  i^  of  it  ? 

649.  A  person  sets  out  at  the  rate  of  11  miles  in  5  hours; 
8  hours  after,  another  person  sets  out  from  the  same  place, 
and  goes  after  him  at  the  rate  of  13  miles  in  3  hours.  How 
far  must  the  latter  travel  to  overtake  the  former  ? 

650.  A  certain  number  of  persons  were  divided  into  three 
classes,  such  that  the  majority  of  the  first  and  second  to- 
gether over  the  third  was  10  less  than  4  times  the  majority 
of  the  second  and  third  together  over  the  first;  but  if  the 
first  had  30  more,  and  the  second  and  third  together  29 
less,  the  first  would  have  outnumbered  the  last  two  by  1. 
The  whole  number  was  34  more  than  8  times  the  majority 
of  the  third  over  the  second.  Find  the  number  in  each 
class. 

651.  A  person  has  a  certain  sum,  half  of  which  he  lends 
at  5  per  cent  interest,  and  half  at  4^  per  cent  interest. 
The  first  loan  yields  him  $60  more  interest  than  the  other. 
What  is  the  amount  of  his  capital  ? 

652.  From  the  two  formulas  in  simple  interest,  a  =  p~\-  i 
and  i  =  trp,  find  what  p  equals  in  terms  of  a,  t,  and  r. 
Express  in  words  the  truth  you  have  expressed  in  letters. 
Make  an  application  of  this  truth  in  solving  a  problem  with 
numbers  in  simple  interest. 

653.  The  width  of  a  rectangular  garden  is  2  rods  less 
than  its  length,  and  |  of  its  area   is  equivalent  to  8  rods 


d:14:  ALGEBRA. 

less  than  |  of  the  square  of  its  width.    How  many  rods  long 
and  wide  is  the  garden  ? 

654.  If  the  side  of  a  square  wei-e  5360  centimeters  longer 
it  would  contain  65536  square  metres.  What  is  the  length 
of  a  side  ? 

655.  Thirty  feet  more  than  ^  of  the  present  height  of  the 
highest  of  the  Egyptian  pyramids  is  equal  to  \  of  its  original 
height.  Three  hundred  feet  less  than  6  times  its  present 
height  is  equal  to  5  times  its  original  height.  Find  the 
original  and  the  present  height. 

656.  One  of  the  parallel  sides  of  a  trapezoid  is  3  inches 
longer  than  the  other;  and  the  altitude  is  1  inch  greater 
than  the  shorter  of  these.  The  area  is  68  square  inches. 
Find  the  altitude  and  the  length  of  each  base. 

657.  The  product  of  the  sum  and  difference  of  two  num- 
bers is  a,  and  the  product  of  the  sum  of  their  squares  by 
the  difference  of  their  squares  is  7na,     Find  the  numbers. 

658.  How  long  will  it  be  before  the  hands  of  the  clock 
will  again  assume  the  same  relative  position  that  they  have 
at  this  moment  ? 

659.  On  a  certain  street  railway  two  sizes  of  cars  are 
used.  What  is  the  seating  capacity  of  each,  if  14  more 
persons  can  be  seated  in  3  large  cars  than  in  4  small  ones, 
and  2  more  persons  in  2  large  cars  than  in  3  small 
ones? 

660.  A  rectangular  lawn  20m.  5  dm.  long  and  8m.  5dm. 
wide  has  a  path  of  uniform  width  around  it.  If  the  area 
of  the  path  equals  62  ca.,  what  is  its  width  ? 

661.  A  takes  3  hours  longer  than  B  to  walk  30  miles; 
but  if  he  doubles  his  pace  he  takes  2  hours  less  time  than 
B.    Find  their  respective  rates  of  walking. 

662.  Divide  20  into  three  parts  such  that  the  products 
of  the  three  pairs  may  be  in  the  ratios  6 :  10 :  15. 

663.  The  first  digit  of  a  number  is  3  times  the  second ;  and 


SUPPLEMENTARY  PROBLEMS.  415 

if  the  number,  increased  by  3,  be  divided  by  the  difference 
of  its  digits,  the  quotient  is  16.     Eequired  the  number. 

664.  Find  the  number  whose  cube  root  is  \  of  its  square 
root. 

665.  A  and  B  can  do  a  piece  of  work  together  in  8  days. 
A  works  alone  4  days,  and  then  both  finish  it  in  5  days 
more.     In  what  time  could  each  have  done  it  alone  ? 

666.  The  sum  of  two  numbers  is  16,  and  the  sum  of  their 
reciprocals  is  \.     What  are  the  numbers  ? 

667.  Find  two  numbers  such  that  their  product,  their 
sum,  and  the  difference  of  their  squares  shall  be  equal  to 
each  other. 

668.  A  certain  number  of  two  digits  is  equal  to  twice  the 
Bum  of  its  digits,  and  the  number  got  by  interchanging  the 
digits  is  equal  to  the  sauare  of  the  sum  of  the  digits.  Find 
the  number. 

669.  Find  three  numbers  of  which  the  first  is  greater 
than  the  second  by  as  many  units  as  the  second  is  greater 
than  the  third*  the  sum  of  the  squares  of  the  three  being 
66. 

670.  Find  two  numbers  whose  product  is  78,  such  that  if 
one  be  divided  by  the  other  the  quotient  is  2  and  the  re- 
mainder 1. 

671.  Divide  1152  into  three  parts,  such  that  9  times  the 
sum  of  the  first  and  second  shall  be  equal  to  7  times  the 
sum  of  the  second  and  third;  and  if  8  times  the  first  be 
subtracted  from  8  times  the  second,  the  remainder  shall  be 
equal  to  the  sum  of  the  first  and  third. 

672.  Find  the  number  whose  square  added  to  its  cube  is 
9  times  the  next  higher  number. 

673.  Two  travellers  set  out  from  two  distant  towns  and 
go  towards  each  other.  When  they  meet  they  find  that  one 
of  them  has  gone  30  miles  more  than  the  other,  and  can 
complete  his  journey  to  the  other  town  in  4  days,  while  the 


416  ALGEBRA, 

other  will  need  9  days  to  complete  his  journey.     How  far 
apart  are  those  two  towns  ? 

674.  A  cask  contains  12  gallons  of  wine  and  18  gallons 
of  water,  and  another  cask  contains  9  gallons  of  wine  and  3 
gallons  of  water;  how  many  gallons  must  be  drawn  from 
each  cask  so  as  to  produce  by  their  mixture  7  gallons  of 
wine  and  7  gallons  of  water  ? 

675.  A  deer  fleeing  from  a  tiger  gets  60  jumps  away  from 
a  certain  tree  before  the  tiger  passes  that  tree;  she  makes 
6  jumps  while  the  tiger  makes  5,  but  7  of  the  tiger's  jumps 
are  equal  to  9  of  the  deer's.  How  many  jumps  from  the 
tree  will  the  tiger  go  before  he  catches  the  deer  ? 

676.  Two  workmen  were  employed  at  different  wages  and 
paid  at  the  end  of  a  certain  time.  The  first,  who  had 
worked  all  the  time,  received  $26.25,  and  the  second,  who 
worked  6  days  less,  received  $19.80.  If  the  second  had 
worked  all  the  time,  and  the  first  had  omitted  six  days, 
their  wages,  taken  together,  would  have  amounted  to 
$48.75.  How  many  days  did  each  work,  and  what  were 
the  wages  of  each  ? 

677.  A  dealer  sells  two  kinds  of  goods,  8  yards  more  of 
the  second  kind  than  of  the  first,  and  receives  $100  from 
the  sale.  He  then  finds  that  he  has  left  just  as  much  of  the 
first  kind  as  he  has  sold  of  the  second,  and  that  the  quantity 
of  the  first  kind  thus  left  is  worth  $100;  and  that  he  has 
left  of  the  second  kind  just  as  much  as  he  has  sold  of  the 
first,  and  that  the  quantity  left  of  the  second  kind  is  worth 
$16.  Find  the  number  of  yards  sold  of  each  kind,  and  the 
price  of  each  per  yard. 

678.  Two  horsemen  start  at  the  same  time,  on  the  same 
road,  from  two  places  15  miles  apart.  At  the  end  of  10 
hours  the  second  horseman  overtakes  the  first,  and  on  com- 
paring their  rates  they  find  that  there  has  been  a  difference 


SUPPLEMENTARY  PROBLEMS,  417 

of  5  minutes  in  the  time  of  going  every  7  miles.  Find  their 
rates,  and  the  distances  they  have  gone. 

679.  Two  pipes  which  supply  the  same  reservoir  fill  it 
in  4  hours  and  12  minutes  when  both  run  together;  but  the 
first  pipe  alone  can  fill  it  in  one  hour  less  than  half  the  time 
in  which  the  second  pipe  alone  can  fill  it.  Find  the  time 
for  each  pipe  alone. 

680.  Find  a  number  such  that  if  it  be  multiplied  by  4 
and  the  product  increased  by  3,  the  result  shall  be  the  same 
as  if  it  were  increased  by  4  and  the  sum  multiplied  by  3. 

681.  Compare  the  rates  of  two  boat^s  crews,  one  rowing 
10  miles  down-stream  in  an  hour,  the  other  5  miles  down 
and  5  miles  back  again  in  an  hour;  supposing  the  current 
to  run  2  miles  an  hour. 

682.  A  sets  off  from  Boston  to  New  York,  and  B  at  the 
same  time  from  I^ew  York  to  Boston,  and  they  travel  uni- 
formly; after  they  have  met  on  the  way,  it  takes  A  16 
hours  to  reach  New  York,  and  B  36  hours  to  reach  Boston. 
Find  in  what  time  each  performed  the  journey. 

683.  A  man  hires  a  certain  number  of  acres  of  land  for 
$336.  He  cultivates  7  acres  for  himself,  and,  by  letting 
the  rest  for  $4  an  acre  more  than  he  paid  for  it,  gets  his 
own  patch  rent  free.     Find  the  number  of  acres  he  hired. 

684.  A  laborer  having  built  105  rods  of  stone  wall  found 
that  if  he  had  built  2  rods  less  a  day  he  would  have  been  6 
days  longer  in  completing  the  job.  How  many  rods  a  day 
did  he  build  ? 

685.  A  and  B  have  together  f  as  much  money  as  C;  B 
and  C  have  together  6  times  as  much  as  A;  and  B  has  $680 
less  than  A  and  C  have  together.     How  much  has  each  ? 

686.  A  certain  man  has  $1400,  which  he  separates  into 
two  portions,  and  puts  at  interest  at  different  rates,  but  so 
that  the  two  portions  produce  equal  returns.  If  the  first 
portion  had  been  lent  at  the  second  rate  of  interest,  it  would 


418  ALomnA. 

have  brought  in  $18  per  annum;  and  if  the  second  portion 
had  been  lent  at  the  first  rate,  it  would  have  brought  in 
132  per  annum.  Find  the  rates  of  interest  and  the  two 
portions  of  the  principal. 

687.  On  a  certain  road  the  telegraph-poles  are  placed  at 
equal  intervals,  and  their  number  is  such  that  if  that  num- 
ber were  less  by  one,  each  interval  between  two  poles  would 
be  increased  by  2||  yards.  Find  the  number  of  poles  and 
the  number  of  intervals  in  a  mile. 

688.  Two  workmen,  A  and  B,  working  together  on  the 
same  job,  complete  it  in  15  days.  It  is  found,  on  the  com- 
parison of  their  work,  that  A,  if  working  alone,  would  have 
required  16  more  days  for  doing  the  whole  job  than  B  would 
have  needed  for  it,  working  alone.  Find  the  time  in  which 
each  workman  would  have  done  the  job  if  working  alone. 

689.  A  certain  share  of  the  profits  of  a  farm  is  divided 
equally  every  month  among  the  hands.  In  July  the  num- 
ber of  hands  was  8  more  than  in  June,  and  each  hand  re- 
ceived $6  less;  the  sum  divided  being  $700.  In  August 
the  number  of  hands  was  2  less  than  in  June,  and  each 
hand  received  $1  less ;  the  sum  divided  being  $540.  Find 
the  sum  divided  in  June. 

690.  A  certain  cistern  is  supplied  by  three  pipes,  A,  B, 
and  C,  of  which  A,  when  running,  discharges  into  the  cistern 
1000  gallons  per  hour.  Pipe  B,  running  alone,  would  fill  the 
cistern  in  8  hours  less  than  A  alone;  pipe  C,  running  alone, 
would  require  5  hours  more  than  A  alone;  while  A  and  B, 
running  together,  would  fill  the  cistern  in  -^-^  the  time  re- 
quired by  C  alone.  Find  the  number  of  gallons  which  the 
cistern  is  capable  of  holding. 

691.  Two  men,  A  and  B,  had  a  money-box,  containing 
$210,  from  which  each  drew  a  certain  sum  daily;  this  sum 
being  fixed  for  each,  but  different  for  the  two.  After  six 
weeks  the  box  was  empty.      Find  the  sum  which  each  man 


StlPPLBMBNTAMT  PROBLEMS,  419 

drew  daily  from  the  box ;  knowing  that  A  alone  would  have 
emptied  it  5  weeks  earlier  than  B  alone. 

692.  A  party  of  friends  went  on  a  pleasure  excursion,  the 
expense  of  which  they  shared  equally.  If  the  number  of  the 
party  had  been  decreased  by  1,  and  if  the  total  expense  had 
been  $150,  the  assessment  for  each  person  would  have  been 
$1  more  than  it  was;  but  if  the  number  of  the  party  had 
been  increased  by  8,  and  if  the  total  expense  had  been  $160, 
the  assessment  for  each  person  would  have  been  $1  less  than 
it  was.  Find  the  number  of  the  party,  tlie  assessment  for 
each  person,  and  the  total  expense  of  the  excursion. 

693.  A  battalion  of  soldiers,  when  formed  into  a  solid 
square,  present  16  men  fewer  in  the  front  than  they  do 
when  formed  in  a  hollow  square  4  deep.  How  many  men 
are  there  in  the  battalion  ? 

694.  Two  mowers,  working  steadily  together,  mow  a  cer- 
tain field  in  3  hours.  On  comparing  their  several  shares  of 
the  work  done,  they  find  that  A  would  have  required,  to  do 
B^s  share,  2|^  hours  more  than  B  would  have  required  to  do 
A^s  share.  Find  the  time  in  which  each  mower,  working 
alone,  would  have  mowed  the  whole  field. 

695.  Two  workmen,  A  and  B,  are  employed  on  a  certain 
job  at  different  wages.  When  the  job  is  finished,  A  receives 
$27,  and  B,  who  has  worked  3  days  less,  receives  $18.75. 
If  B  had  worked  for  the  whole  time,  and  A  3  days  less  than 
the  whole  time,  they  would  have  been  entitled  to  equal 
amounts.  Find  the  number  of  days  each  has  worked,  and 
the  pay  each  receives  per  diem. 

696.  Several  friends  on  an  excursion  spent  a  certain  sum 
of  money.  If  there  had  been  5  more  persons  in  the  party, 
and  each  person  had  spent  25  cents  more,  the  bill  would 
have  amounted  to  $33.  If  there  had  been  2  less  in  the 
party,  and  each  person  had  spent  30  cents  less,  the  bill 


420  ALGEBRA, 

would  have  amounted  to  only  $11.     Of  how  many  did  the 
party  consist,  and  what  did  they  spend  ? 

697.  A  man  walks  2  hours  at  the  rate  of  4|-  miles  per 
hour;  he  then  adopts  a  different  rate.  At  the  end  of  a 
certain  time  he  finds  that  if  he  had  kept  on  at  the  rate  at 
which  he  had  set  out,  he  would  have  gone  three  miles 
further  from  his  starting-point;  and  that  if  he  had  walked 
3  hours  at  his  first  rate  and  |-  hour  at  his  second  rate,  he 
would  have  reached  the  point  he  has  actually  attained. 
Find  the  whole  time  occupied  by  the  walk,  and  his  final 
distance  from  the  starting-point. 

698.  A  reservoir,  supplied  by  several  pipes,  can  be  filled 
in  15  hours,  every  pipe  discharging  into  it  the  same  fixed 
number  of  hhds.  per  hour.  If  there  were  5  more  pipes, 
and  every  pipe  discharged  per  hour  7  hhds.  less,  the  reser- 
voir would  be  filled  in  12  hours.  If  the  number  of  pipes 
were  1  less,  and  every  pipe  discharged  per  hour  8  hhds. 
more,  the  reservoir  would  be  filled  in  14  hours.  Find 
number  of  pipes  and  capacity  of  reservoir. 

699.  A  man  bought  a  certain  number  of  railway  shares 
when  they  were  at  a  certain  rate  per  cent  discount  for 
$8500;  and  afterwards,  when  they  were  at  the  same  rate 
per  cent  premium,  he  sold  all  but  20  of  them  for  $9200. 
How  many  did  he  buy,  and  what  did  he  give  for  each  of 
them  ? 

700.  A  and  B  can  do  a  piece  of  work  in  18  days;  A 
and  0  can  do  it  in  45  days ;  B  and  C  in  20  days.  Find 
the  time  in  which  A,  B,  and  C  can  do  it,  working  together. 

701.  A  man  bought  a  certain  number  of  sheep  for  $300; 
he  kept  15  sheep,  and  sold  the  remainder  for  $270,  gaining 
half  a  dollar  a  head.  How  many  sheep  did  he  buy,  and  at 
what  price  ? 

702.  A  hires  a  certain  number  of  acres  for  $420.  He 
lets  all  but  4  of  them  to  B,  receiving  for  each  acre  $2.50 


SUPPLEMENTARY  PROBLEMS,  421 

more  than  he  paid  for  it.     The  whole  amount  received 
from  B  is  $420.     Find  the  number  of  acres. 

703.  A  man  walks  at  a  regular  rate  of  speed  on  a  road 
which  passes  over  a  certain  bridge  distant  21  miles  from 
the  point  the  man  has  reached  at  noon.  If  his  rate  of 
speed  were  half  a  mile  per  hour  greater  than  it  is,  the  time 
at  which  he  crosses  the  bridge  would  be  an  hour  earlier 
than  it  is.  Find  his  actual  rate  of  speed  and  the  time  at 
which  he  crosses  the  bridge      Explain  the  negative  answer. 

704.  A  man  setting  out  on  a  journey  drives  at  the  rate 
of  a  miles  per  hour  to  the  nearest  railway  station,  distant 
1)  miles  from  his  house.  On  arriving  at  the  station  he  finds 
that  the  express  for  his  destination  has  left  c  hours  before. 
At  what  rate  should  he  have  driven  to  catch  the  express  ? 
Having  obtained  the  general  solution,  find  what  the  an- 
swer becomes  in  the  following  cases; 

(1)0=0;     {2)c=J-;     (3)  ^  =  -  ^. 

In  case  (2)  how  much  time  does  the  man  have  to  drive 
from  his  house  ?  In  case  (3)  what  is  the  meaning  of  the 
negative  value  of  c? 

705.  A  landowner  laid  out  a  rectangular  lot  containing 
1200  square  yards.  He  afterwards  added  3  yards  to  one 
dimension  of  his  lot,  and  subtracted  1^  yards  from  the 
other,  thereby  increasing  the  area  of  his  lot  by  60  square 
yards.  Find  the  dimensions  of  the  lot  before  and  after 
the  change.     How  do  you  explain  the  negative  answer  ? 

706.  A  vessel  is  half  full  of  a  mixture  of  wine  and 
water.  If  filled  up  with  wine,  the  ratio  of  the  quantity  of 
wine  to  the  quantity  of  water  is  10  times  what  it  would  be 
if  the  vessel  were  filled  up  with  water.  Find  the  ratio  of 
the  original  quantity  of  wine  to  that  of  water. 

707.  Three  students,  A,  B,  and  0,  agree  to  work  out  a 


422  ALOEBBA. 

series  of  difficult  problems,  in  preparation  for  an  examina- 
tion; and  each  student  determines  to  solve  a  fixed  number 
every  day.  A  solves  9  problems  per  day,  and  finishes  the 
series  4  days  before  B;  B  solves  2  more  problems  per  day 
than  C,  and  finishes  the  series  6  days  before  C.  Find  the 
number  of  problems,  and  the  number  of  days  given  to  them 
by  each  student. 

708.  A  certain  whole  number  composed  of  three  digits 
has  the  following  properties:  10  times  the  middle  digit 
exceeds  the  square  of  half  the  sum  of  the  digits  by  21; 
if  99  be  added  to  the  number,  the  order  of  the  digits  is 
inverted;  and  if  the  number  be  divided  by  11,  the  quotient 
is  a  whole  number  of  two  digits,  which  are  the  same  as  the 
first  and  last  digits  of  the  original  number.  Find  the 
number. 

709.  A  certain  manuscript  is  divided  between  A  and  B 
to  be  copied.  At  A^s  rate  of  work,  he  would  copy  the 
whole  manuscript  in  18  hours;  B  copies  9  pages  per  hour. 
A  finishes  his  portion  in  as  many  hours  as  he  copies  pages 
per  hour;  B  is  occupied  2  hours  more  than  A  upon  his 
portion.  Find  the  number  of  pages  in  the  manuscript, 
and  the  number  of  pages  in  the  two  portions. 

710.  Two  casks,  of  which  the  capacities  are  in  the  ratio 
of  a  to  h,  are  filled  with  mixtures  of  water  and  alcohol. 
If  the  ratio  of  water  to  alcohol  is  that  of  m  to  n  in  the  first 
cask,  and  that  of  ji?  to  g  in  the  second  cask,  what  will  be 
the  ratio  of  water  to  alcohol  in  a  mixture  composed  of  the 
whole  contents  of  the  two  casks  ?  Eeduce  the  answer  to 
its  simplest  form.  What  does  the  answer  (in  its  simplest 
form)  become  if  m  =  q  =  0  ]  and  what  is  the  simplest 
statement  of  the  question  in  this  case  ? 

711.  A  boat's  crew,  rowing  at  half  their  usual  speed,  row 
3  miles  down  a  certain  river  and  back  again,  in  the  middle 
of  the  stream,  accomplishing  the  whole  distance  in  2  hours 


SUPPLEMENTAUY  PROBLEMS.  423 

and  40  minutes.  Find  (in  miles  per  hour)  the  rate  of  the 
crew  when  rowing  at  full  speed,  and  the  rate  of  the  current. 
(Notice  both  solutions  of  this  problem.) 

712.  A  and  B  have  4800  circulars  to  stamp  for  the  mail, 
and  mean  to  do  them  in  2  days,  2400  each  day.  The  first 
day.  A,  working  alone,  stamps  800  circulars,  and  then  A 
and  B  together  stamp  the  remaining  1600;  the  whole  job 
occupying  3  hours.  The  second  day  A  works  3  hours,  and 
B  1  hour;  but  they  accomplish  only  y\-  of  their  task  for  that 
day.  Find  the  number  of  circulars  which  each  stamps  per 
minute,  and  the  length  of  time  that  B  works  on  the  first 
day. 

713.  A  broker  sells  certain  railway  shares  for  13240.  A 
few  days  later,  the  price  having  fallen  $9  per  share,  he  buys 
for  the  same  sum  5  more  shares  than  he  had  sold.  Find 
the  price,  and  the  number  of  shares  transferred  each  day. 

714.  At  6  o'clock  on  a  certain  morning,  A  and  B  set  out 
on  their  bicycles  from  the  same  place,  A  going  north  and  B 
south,  to  ride  until  1:30  p.m.  A  moved  constantly  north- 
wards at  the  rate  of  6  miles  per  hour.  B  also  moved  always 
at  a  fixed  rate;  but  after  a  while  he  turned  back  to  join  A. 
Four  hours  after  he  turned,  B  passed  the  point  at  which  A 
was  when  B  turned;  and  at  1:30  p.m.,  when  he  stopped, 
he  had  reduced,  by  one-half,  the  distance  that  was  between 
them  at  the  time  of  turning.  Find  B's  rate,  the  time  at 
which  he  turned,  the  distance  between  A  and  B  at  that 
time,  and  the  time  at  which  B  would  have  joined  A  if  the 
ride  had  been  continued  at  the  same  rates  of  speed.  Find 
the  answers  for  both  solutiofis. 

715.  Two  travellers,  A  and  B,  go  from  P  to  Q  at  uni- 
form but  unequal  rates  of  speed.  A  sets  out  first,  travel- 
ling on  foot  at  the  rate  of  20  minutes  for  every  mile.     B 

PQ 

follows,  going  1  mile  while  A  traverses  the  distance  -— .   B 

oO 


424  ALGEBRA. 

overtakes  and  passes  A,  8  miles  from  P;  and  when  B 
reaches  Q,  he  is  9  miles  ahead  of  A.  Find  the  distance 
PQ,  and  B's  rate  of  speed  in  minutes  to  the  mile. 

716.  Tristram  is  10  years  younger  than  Launcelot;  and 
the  product  of  the  ages  they  attained  in  1870  is  96.  Find 
the  ages  they  attain  in  1888. 

717.  A  certain  librarian  spends  every  year  a  fixed  sum 
for  books.  In  1886,  the  cost  of  his  purchases  averaged  $2 
per  volume;  in  1887,  he  bought  300  more  volumes  than  in 
1886;  and  in  1888,  300  more  volumes  than  in  1887.  The 
average  cost  per  volume  was  30  cents  lower  in  1888  than 
in  1887.  Find  the  number  of  volumes  bought  each  year, 
and  the  fixed  price  paid  for  them. 

718.  Two  tanks,  A  and  B,  are  discharging  water;  A  at 
the  rate  of  x  barrels  per  hour,  and  B  at  the  rate  oi  x  -\-100 
barrels  per  hour.  At  a  time  {1  -\-  y)  hours  after  noon,  A 
contains  480  barrels  less  than  at  noon ;  and  at  a  time  {1  —  y) 
hours  after  noon,  B  contains  400  barrels  less  than  at  noon. 
Find  the  rate  at  which  eacli  tank  is  discharging  water ;  and 
the  times  {l-\-  y)  and  (1  —  y)  hours  after  noon. 

719.  A  certain  railway  runs  due  east  and  west,  P,  Q  and 
E  being  successive  stations  on  the  road  from  east  to  west, 
and  so  situated  that  PQ  =  216  miles,  PR  =  240  miles.  Two 
trains,  on  parallel  tracks,  pass  P  simultaneously  at  noon. 
The  rate  of  motion  of  train  No.  2  from  east  to  west  is  8 
miles  per  hour  less  than  that  of  train  Iso.  1 ;  and  train  No. 
2  passes  Q  1|  hours  later  than  train  No.  1  passes  E.  Find 
the  rate  and  direction  of  motion  of  each  train ;  and  find  the 
hour  at  which  train  No.  1  passes  E,  and  the  hour  at  which 
train  No.  2  passes  Q. 

720.  Two  wheelmen,  A  and  B,  are  riding  eastward  over 
the  same  road,  B's  eastward  rate  exceeding  A^s  by  4  miles 
an  hour.  A  certain  milestone,  M,  is  passed  by  A  at  noon, 
and  by  B  15  minutes  later;  a  second  milestone,  N,  6  miles 


SUPPLEMENTAET  PROBLEMS.  425 

east  of  M,  is  passed  by  both  wheelmen  at  the  same  instant. 
Find  the  rate  of  each  wheelman  in  miles  per  hour,  and  the 
time  of  their  passing  N^. 

721.  A  and  B  start  at  the  same  time  from  two  towns 
and  travel  towards  each  other.  When  they  meet,  B  has 
travelled  a  miles  more  than  A;  and  it  will  take  A  d  hours 
longer,  and  B  c  hours  longer,  for  each  to  reach  the  town 
the  other  has  left.     Find  the  distance  between  the  towns. 

722.  A  gentleman  has  two  horses  and  one  chaise.  The 
first  horse  is  worth  a  dollars  less,  and  the  second  horse  h 
dollars  less,  than  the  chaise.  If  f  of  the  value  of  the  first 
horse  be  subtracted  from  that  of  the  chaise,  the  remainder 
will  be  the  same  as  if  J  of  the  value  of  the  second  horse  is 
subtracted  from  twice  that  of  the  chaise.  Find  the  value 
of  each  horse  and  that  of  the  chaise.  What  are  the 
answers  ii  a  =  —  50;  ^  =  50  ? 

723.  Divide  the  number  a  into  two  such  parts  that  if 
the  first  be  multiplied  by  m  and  the  second  by  7i,  the  sum 
of  the  products  is  d.  In  what  case  would  the  terms  of  the 
fractional  values  of  the  unknown  quantities  become  zero  ? 
and  how  could  they,  then,  satisfy  the  conditions  of  the 
problem  ? 

724.  A  courier  started  from  a  certain  place  n  days  ago, 
and  makes  a  miles  daily.  He  is  pursued  to-day  by  another 
making  Z>  miles  daily.  In  how  many  days  will  the  second 
overtake  the  first  ?  In  what  case  would  both  the  terms 
of  the  fractional  value  of  the  unknown  quantity  become 
zero  ?  and  how  could  this  value  be  a  solution  ? 

725.  A  wine  merchant  has  two  kinds  of  wine:  the  one 
costs  a  shillings  per  gallon,  the  other  1)  shillings.  How 
must  he  mix  both  these  wines  together  in  order  to  have  n 
gallons  worth  c  shillings  per  gallon  ?  In  what  cases  would 
the  values  of  either  of  the  unknown  quantities  be  nega- 
tive ?     Why  should  this  be  the  oase,  and  could  the  enun- 


426  ALGEBRA. 

elation  be  eorrected  for  this  ease  ?  In  what  eases  would 
the  value  of  one  of  the  unknown  quantities  be  zero^  and 
what  would  this  value  signify  ? 

726.  The  owners  of  a  certain  mill  make  a  dollars  a  day 
each,  sharing  equally.  If  the  number  of  owners  were  h 
less,  they  would  make  c  dollars  each.  Eequired  the  number 
of  owners  and  the  total  daily  profit.  What  are  the  answers 
if  «  =  80,  ^  =  -  3,  c  =  50  ? 

727.  Two  trains  are  running  westward  on  parallel  tracks, 
at  the  rate  of  a  and  l  miles  an  hour  respectively.  At  noon 
they  are  m  miles  apart.  When  are  they  together  ?  When 
is  the  unknown  quantity  positive  ?  When  does  it  become 
zero  ?  infinite  ?  indeterminate  ?  negative  ?  Explain  the 
signification  of  each  result. 

728.  A  certain  sum  of  money  will  amount  to  a  dollars  in 
m  months,  and  to  h  dollars  in  n  months.  Find  the  princi- 
pal and  the  rate  of  interest.  Find  the  answers  when 
a  =  1837.50,  b  =  1890,  m  =  10,  n  =  16. 

729.  A  banker  has  two  kinds  of  coin.  It  takes  a  pieces 
of  the  first,  or  b  pieces  of  the  second,  to  make  a  dollar.  If 
a  dollar  is  offered  for  c  pieces,  how  many  of  each  kind  must 
be  given  ? 

730.  I  have  a  counters  distributed  unequally  among 
three  cups.  Taking  some  of  the  counters  from  the  first 
cup,  I  double  the  number  of  counters  in  the  second  and 
third ;  next,  taking  from  the  second,  I  double  the  number 
in  the  first  and  third;  and  lastly,  taking  from  the  third 
cup,  I  double  the  numbers  in  the  first  and  second.  Then 
I  find  that  the  second  cup  contains  twice,  and  the  third 
cup  three  times,  as  many  counters  as  the  first.  How  many 
in  each  cup  at  the  beginning  ? 

731.  A  merchant  who  had  two  brands  of  flour  sold  a 
barrels  of  the  first  and  b  barrels  of  the  second  at  an  average 
price  of  $c  per  barrel;  and  at  the  same  rates,  he  sold  m 


SUFPLEMEl^TAET  PROBLEMS,  427 

barrels  of  the  first  and  n  barrels  of  the  second  at  an  aver- 
age price  of  $^  per  barrel.     Find  the  price  of  each  brand. 

732.  A  person  has  a  hours  at  his  disposal.  How  far  can 
he  ride  in  a  coach  at  the  rate  of  b  miles  an  hour,  so  as  to 
get  back  in  time,  walking  at  the  rate  of  c  miles  an  hour  ? 

733.  A  cask  of  wine  contains  a  gallons;  b  gallons  are 
drawn  off,  and  the  cask  filled  up  with  water.  After  this 
has  been  done  n  times,  how  many  gallons  of  the  original 
wine  are  in  the  cask  ? 


INDEX. 


Abbreviation  of  explanations,  1 

of  rules,  41 
Addition,  61 

see  also  Summation. 
Age  problem,  12 
Algebra,    three    uses   of,   1,   41, 

88 
Algebraic  scale,  132 
Alternation,  300 
Analogy,  303 

between  terms  and  factors,  241 
Answers  alike,  107 

apparently  different,  108 

infinite  list  of,  135 

interpreted,  110,  285 

meaningless,  115 

suggest  related  problems,  113 

tabulated,  172 

to  quadratic  pairs,  170 
Antecedents,  299 
Approximate  square  roots,  52 
Arithmetic  complement,  354 
Arithmetic  mean,  318 
Arithmetic  progression,  310 
Arrangement   in    multiplication, 

81 
Associative  law,  59,  60,  72 
Augmented  logarithms.  353 
Auxiliary  quadratic,  182 
Axes,  coordinate,  133 
Axiom,  15 
Axiom  A,  104 
Base,  349 

Binary  equation,  174,  176 
Binomial,  70 

formula,  341,  343 

quadratic  surd,  256 
theorem,  334,  344 


Binomial  theorem,  any  one  term, 

340 
Boundary  between  +  and  —  num- 
bers, 287 
Brackets,  see  Parentheses. 
Calculation  by  logarithms,  353 
Cautions  for  H.  C.  F.,  211 
Changing  signs,  63,  241 

see  also  Law  of  signs. 
Characteristic,  351 
Circular  race  problem,  39 
Circulating  decimal,  321 
Cistern  problem,  36.  289 
Clearing  of  fractions,  218 
Clock  problem,  38 
Coefficients,  15 

detached,  328 

irregular,  337 

of  powers,  333 

symmetrical,  336 
Coin  and  bill  problem,  5 
Cologarithms,  354 
Combination,  141,  170 
Common  difference,  310 

logarithms,  349 

ratio,  315 

roots,  205 
Commutative  law,  59,  60 
Comparison,  185 
Complete  list  of  answers,  135 
Completing  the  square,  96 
Complex  binomials,  258 

fraction,  230 
Composition,  301 

and  division,  301 
Compound  interest,  55 
Conditions,  88,  137,  151,  153 
Conjugate  surds,  252 

429 


i30 


INDEX, 


Consequent,  299 
Consistent  equations,  151 
Continued  equation,  155 

fractions,  232 

multiplication,  69 

proportion,  305 
Conventions,  4 
Convergent  series,  321 
Corresponding  values,  131,  306 
Cross-multiplication,  77 
Cross-products,  77 
Cube  roots,  267,  270 
Current  problem,  25 
Cyclic  order,  221 
Z>,  350 

d,  350 

Day's  work  problem,  36 
Degree,  166 

Demonstration  of  theorems,  88 
Detached  coefficients,  328 
Diagrams   for   linear    equations, 
133-136 

for  quadratics,  138,  169 
Difference  of  like  powers,  Ex.  11, 
p.  198 

of  the  cubes,  90 

of  the  squares,  89 

of  unknown  numbers  given,  8 
Digit  problem,  22 
Discussion  of  equations,  292 

of  problems,  285 

of  the  quadratic,  294 
Distributive  factoring,  73 

law,  11,  72,  248 
Divergent  series,  321 
Dividend,  82 
Division,  83,  194 

(in  proportion),  301 
Divisor,  82 
Double -cross-products,  97 

e,  359 
Elimination,  141 

by  combination,  141 

by  comparison,  185 

by  detached  coefficients,  332 

by  substitution,  166 

generalized,  292 

finding  a  binary,  176 

first  method,  131 

for  H.  C.  F.,  200,  203 

irregular  devices,  183 

more  than  three  letters,  158 


Elimination,     numerical     terms, 
176 

second  method,  163 

three  letters,  153 

where  it  fails,  150 
Entire  surd,  246 
Equal  factors,  69 
Equation,  3 

binary,  174 

continued,  155 

degree  of,  166 

factorable,  104 

homogeneous,  174 

how  to  form,  405 

identical,  88 

of  condition,  88 

simple,  rule  for,  15 

with  radicals,  258 
Equations,  inconsistent,  151 

independent,  151 

linear  and  quadratic,  163 

of  the  New  Set,  154 

solved  like  quadratics,  262 

vs,  expressions,  217 
Equilateral  triangle,  42 
Equiradical  surds,  246 
Expansion,  335 
Exponent,  see  Index 
Expression,  3,  217 
Extremes,  299 
Factor,  69 

Factorable  equations,  104 
Factorial,  338 

Factoring  by  completing  square, 
99 

by  cross-multiplication,  79 

by  parts,  126 

by  theorem  A ,  93 

distributive,  73 

for  the  L.  C.  M.,  215 

higher  equations,  125 

literal  quadratics,  278 

quadratic  pairs,  173 
Factors  of  an  equation,  104 

test  for  simple,  197,  235 
Falling  bodies,  42 
First  method  of  elimination,  131 

use  of  algebra,  1 
Formula,  41 

binomial,  341 

for  area  of  triangle,  50 

for  in -radius,  356 


INDEX. 


431 


Formula  for  interest,  54 
for  solving  quadratics,  122 
for  square  root,  51 
for  X,  166 

Formulae,  application  of,  46 
for /and  s,  810,  315 
involving  square  root,  53 
translated  into  rules,  41 

Four   answers    (pair  of  quadrat- 
ics), 170 

Four-place  tables,  349 

Fourth  proportional,  304 
roots,  272 

Fractional  equations,  28,  217 
indices,  244,  345,  348 

Fractions,  190 
complex,  230 
continued,  232 
laws  of,  192 

Fundamental  laws,  72 

^,43 

Generalization,  274 
linear  equations,  292 
quadratic  equation,  294 
the  binomial  theorem,  338 

Geometric  mean,  304,  318 
progression,  315 

Graphical  method,  132 

G.  C.  D.,  G.  C.  M.,  see  H.  C.  F. 

Grouping  of  factors,  60 
of  terms,  59 

Harmonic  progression,  323 

H.  C.  F.,  198,  203,  208 
as  related  to  L.  C.  M.,  236 
by  detached  coefficients,  331 
of  three  expressions,  207 

Highest  and  lowest  terms,  201 

History  of  equations,  3 

Homogeneous  equation,  174,  176 

Homologous  terms,  302 

Hyperbola,  170 

i  =  |/^.  118 
Identities,  88 
Imaginaries,  119,  254 
Imaginary  unit,  254 
Impossible  problems,  151.  293 
Inconsistent  equations,  151 
Independent  equations,  151 
Indeterminate  equations,  152 
Index,  69,  239 

distributive  law,  248 

fractional,  244 


Index  negative,  240 

sign  of,  240 

zero,  240 
Induction,  344 
Infinite  geometric  series,  319 
Infinite  list  of  answers,  135 
Infinity,  288 
In-radius,  356 
Interest  formula.  54 
Interpolation,  350 
Intersections,  140 
Inverse  variation,  307 
Inversion,  300 
Irrational  quantity,  177,  246 
Irregular  coefficients,  337 

elimination,  182 
Known    numbers,    symbols   for, 

42,  374 
Law,  Associative,  Commutative, 
and  Distributive,  72,  248 

of  signs,  see  Signs. 
Laws  of  algebra,  72 

of  fractions,  192 

of  imaginaries,  254 

of  indices,  239 

of  proportion,  299 
L.  C.  M  ,  212 

by  factoring,  215 

by  mems  of  H.  C.  F.,  236 

of  factorable  equations,  234 

of  three  expressions,  237 
Least    Common     Multiple,      see 

L.  C.  M. 
Light-year,  858 

Like   powers,    sums   and   differ- 
ences of,  Exs.   11  and  12,  p. 
198 
Limiting  values,  285 
Linear  equation,  166 

and  quadratic,  163,  17i 
Literal  equation,  275 

quadratics,  278,  384 

simultaneous  equations,  281 
Logarithm,  348 
Logarithmic  schedule,  355 
Long  division,  84 
Lowest    Common    Multiple,    see 

L.  C.  M. 
Mantissa,  352 

Mathematical  induction,  344 
Mean  proportional,  304 
Meaning  of  negative  answers,  110 


4:32 


INDEX. 


Meaningless  answers,  115 
Means,  299,  319 
Member  of  an  equation,  3 
Minus,  see  Negative 
Mixed  number,  226 

surd,  246 
Mixture  problem,  19 
Modified  reduction  of  fractions, 

225 
Modulus,  341 
Monomial,  71 

factors,  73 
Multiple,  212 
Multiplication,  69 

distributive  principle,  11 

indicated  without  sign,  3 

of  fractions,  193 

of  polynomials,  75 

of  proportions,  302 
Napierian  logarithms,  359 
Negative  characteristic,  352 

fractions,  33 

index,  240,  345 

quantities,  56 

terms,  57 
Nests  of  parentheses,  66 
New  Set,  154,  158 
New  work  on  old  patterns,  28 
Non-algebraic  conditions,  3  51 
Notation  by  powers  of  10,  358 
Numerical  terms  eliminated,  176 
Order  of  factors,  60 

terms,  59 
Pairs,  linear-quadratic,  163 

of  quadratics,  169 
Parabola,  170 
Parentheses,  34,  64,  67 
Partial  products,  76,  81,  126 
Pattern  expansions,  337 
7f,  value  of,  41 
Polynomials,  71 
Positive  terms,  57 
Power,  69,  239 
Powers,  arrangement  br,  81 

of  10,  348,  358 
Principal  dividing  line.  230 

signs  of  a  fraction,  220 
Problems  put  into  equations,  405 
Products,  77 
Progressions,  310 
Proof  of  binomial  formula,  343 

of  theorems,  88 


Proportion,  299 
Proportional  parts,  351 
Quadratic,  78,  108.  166 

and  linear,  163 

completing  the  square,  99 

expression,  78 

form,  262 

formula,  122 

generalized,  294 

literal,  278 

pairs,  170 

standard  form,  121 

surd,  246 

with  three  letters,  187 

with  zero  terms,  295 
Quotient,  82 
Radical  equations,  258 
Radicals,  246 
Ratio,  191 

Rational  expression,  246 
Rationalizing  a   quadratic  surd 
250 

a  term,  250 

factors,  253 
Rearranging  denominators,  221 

factors,  60 

terms,  59 
Reciprocal,  58 

of  a  power,  240 
Reciprocals   in   fractional   equa- 
tions, 227 

solving  for,  148 
Reduction  of  equations,  11 

of  fractional  expressions,  225 

of  fractions,  193 

of  fractional  products.  194 
Removing  parentheses,  65 
Root,  69,  239 
Roots,  common,  205 

finding,  51,  265,  356 

of  an  equation,  124,  384 

of  proper  fractions,  358 

of  a  quadratic,  294 
Rule  for  addition  in  algebra,  61 

for  literal  equations,  275 

for  simple  equations,  15 

for  subtraction,  63 

of  signs,  see  Signs. 

of  Three,  300 
Rules,  abbreviation  of,  41 

application  of,  44 

translated  into  formulae,  43 


INDEX. 


433 


s,  50 

Scale,  algebraic,  132 
Scales  of  notation,  24,  330 
Schedule,  logarithmic,  355 

of  steps,  3 
Second   method   of   elimination, 

163 
Second  use  of  algebra,  41 
Series,  310 

by  expansion,  345 

convergent,  321 

for  cube  root,  271 

for  square  root,  266 

of  equal  ratios,  303 
Shortages,  12 
Sign  of  identity,  88 
Signs  in  division,  83 

in  multiplication,  76 

in  subtraction,  63 

of  a  fraction,  220 

of  indices,  241 

of  parentheses,  65 
Similar  surds,  247 

terms,  15 
Simple  equation,  rule,  15 
Simplest  form  of  a  surd,  247 
Simplification  due  to  logarithms, 

348 
Simultaneous  equati  ns,  139 

literal  equations,  281 

values,  306 
Solving  simple  equations,  15 
Sphere,  42 
Square  of  a  polynomial,  89 

of  the  difference,  89 

of  the  sum,  89 

root,  51,  265 

root  of  a  binomial  surd,  256 
Squares  by  Theorem  A,  91 
Standard  form,  121 
Straight  products,  77 
Substituting  for  a?,  164 
Substitutions,  86 
Subtraction,  62 
Sum  of  cubes,  90 

of  like  powers,  Ex.  12,  p.  198 

of  unknown  numbers  given,  19 
Summation,  60 

of  fractions,  213 
Supplementary  problems,  366 


Surd,  177,  246 

Symmetrical  elimination,  179 
Symmetry   in    binomial     expan- 
sion, 336 

not  obvious,  184 
Systems  of  logarithms,  359 
Table  of  logarithms,  363 

of  powers  of  10,  348 
Tabular  difference,  850 
Tabulated  answers,  172 
Terms  of  an  expression,  3 
Tests  for  simple  factors,  197,  235 
Theorem  A,  89 
Theorems,  88 

of  fractions,  193 

of  proportion,  299 
Third  method  of  elimination,  185 

use  of  algebra,  88 
Three  equations,  one  quadratic, 
187 

means,  324 

principal  signs,  220 

unknown  letters,  153,  187 

uses  of  algebra,  1 
Transformations,  56 
Translating  formulae  into  rules, 
41 

identities  into  theorems,  90 

rules  into  formula?,  43 

the  sign  — ,  58 
Transposition,  rule  of,  15 
Triangle,  area  of,  50 

in- radius,  356 
Trinomial,  70 

Two  answers    (pair  of   quadrat- 
ics), 172 

(one  quadratic),  104 
Unit,  imaginary,  254 
Unknown  letters,  more  than  two, 
152 

numbers,  symbols  for,  4 
Using  logarithm  tables,  349 
Valuation  problem,  5 
Value  of  It,  41 
Variables,  68,  284,  305 
Variation,  theorem  of,  308 
Verifying  expansions,  345 
Zero,  287 

coefficients,  295,  329 

index,  240 


MATHEMATICS 


Gillet's  Elementary  Algebra. 

By  J.  A.  GiLLET,  Professor  in  the  New  York  Normal  College,  xiv  +  412 
pp.  i2mo.  Half  leather.  $1.10.  With  Part  II,  xvi  +  512  pp.  i2mo. 
$1.35. 

Distinguished  from  the  other  American  text-books  covering 
substantially  the  same  ground,  (i)  in  the  early  introduction 
of  the  equation  and  its  constant  employment  in  the  solution 
of  problems  ;  (2)  in  the  attention  given  to  negative  quantities 
and  to  the  formal  laws  of  algebra,  thus  gaining  in  scientific 
rigor  without  loss  in  simplicity  ;  (3)  in  the  fuller  development 
of  factoring,  and  in  its  use  in  the  solution  of  equations. 


James  L.  Love,  Professor  in 
Harvard  University  : — It  is  un- 
usually good  in  its  arrangement 
and  choice  of  material,  as  well  as 
in  clearness  of  definition  and  ex- 
planation. 

J.  B.  Coit,  Professor  in  Boston 
University : — I  am  pleased  to 
see  that  the  author  has  had  the 
purpose  to  introduce  the  student 
to  the  reason  for  the  methods  of 
algebra,  and  to  avoid  teaching 
that  which  must  be  unlearned 
when  the  student  moves  on  into 
higher  studies. 

William  A.  Francis,  Phillips 
Academy,  Exeter,  N.  H.: — I  find 
many  things  in  it  to  admire,  es- 
pecially the  shortening  of  alge- 
braic processes. 

F.  F.  Thwing,  Manual  Train- 
ing High  School,  Louisville,  Ky.: — 
Two  features  strike  me  as  being 
very  excellent  and  desirable  in  a 
text-book,  the  prominence  given 
to  the  concrete  problems  and  the 
application  of  factoring  to  the  so- 
lution of  quadratic  equations. 

E.  F.  Lohr,  Duluth  {Minn) 
High  School: — What  I  especially 
like  is  the  author's  early  intro- 
duction of  the  equation  and  the 
prominence  given  to  drill  in  prob- 
lems. 


A.  L.  Baker,  Professor  in  Uni- 
versity of  Rochester,  N.  V.: — The 
author's  diagramatic  method  of 
multiplying  and  factoring  should 
be  a  valuable  help  to  the  student. 
His  treatment  of  undetermined 
coefficients  is  particularly  good. 
The  early  introduction  of  syn- 
thetic division  is  to  becommended. 
I  cannot  speak  too  highly  of  his 
treatment  of  negative  quantities. 
It  is  an  excellent  book  in  every 
way,  and  thoroughly  up  to  date 
in  many  little  ways  that  only  a 
practical  teacher  can  appreciate. 

Pomeroy  Ladue,  Professor  in 
New  York  University  : — The  book 
is  an  excellent  one,  free  from 
most  of  the  imperfections  of  many 
American  text-books,  and  very 
well  adapted  for  beginners.  .  .  . 
indicates  that  it  is  written  by  a 
successful  teacher. 

J.  G.  Estill,  Hot ch kiss  School, 
Lakeville,  Conn.: — The  order  in 
which  the  subjects  are  taken  up  is 
the  most  rational  of  any  algebra 
with  which  I  am  familiar. 

Mary  C.  Noyes,  Lake  Erie  Fe- 
male Seminary,  Painesville,  O.: — 
The  book  seems  to  be  particularly 
clear  in  its  statements  and  expla- 
nations. I  have  noticed  particu- 
larly the  explanation  of  the  theory 
of  exponents. 


Mathematics 


Gillet's  Euclidean  Geometry. 

By  J.  A.  GiLLET,    Professor  in  the   New  York  Normal  College.     436  pp. 
i2mo.     Half  leather.     $1.25. 

This  book  is  "  Euclidean  "  in  that  it  reverts  to  purely  geo- 
metrical methods  of  proof,  though  it  attempts  no  literal  repro- 
ductions of  Euclid's  demonstrations  or  propositions.  Metrical 
applications  and  illustrations  of  geometrical  truths  are  inter- 
spersed with  unusual  freedom.  *^  Originals "  are  made  an 
integral  part  of  the  logical  development  of  the  subject  ;  that 
is,  they  include  theorems  and  problems  which  are  ordinarily 
found  in  the  regular  series,  and  thus,  by  devoting  the  pupil's 
inventive  energies  to  these  instead  of  to  merely  incidental 
^'originals,"  he  secures  the  necessary  power  at  a  smaller  ex- 
penditure of  time. 


Percy  F.  Smith,  Professor  in 
Yale  University : — The  return  of 
the  *'  spirit  of  Euclid"  should  be 
much  appreciated,  and  ft  will  be 
interesting  to  watch  the  workings 
in  the  class  room  of  the  two  alter- 
native methods  of  Book  V.  Con- 
sistency and  rigor  are  carefully 
maintained  in  both  works,  and  I 
shall  take  great  pleasure  in  using 
and  recommending  them. 

Dr.  Charles  A.  Pitkin,  Thayer 
A  cadeniy.  South  Braintree^  Mass. : — 
I  find  it  to  be  a  very  attractive 
book,  both  in  its  general  appear- 
ance and  in  the  clearness  with 
which  the  demonstrations  are  set 
forth.  The  somewhat  unusual 
arrangement  of  the  order  in  which 
the  author  gives  the  earlier  prop- 
ositions seems  to  me,  in  general, 
a  good  one.  The  problems  and 
exercises  are  well  selected. 

C.  W.Green,  Northwestern  Uni- 
versity, III.: — The  book  is  of  supe- 
rior merit,  especially  in  the  sim- 
plicity and  completeness  of  its 
demonstrations. 

E.  P.  Thompson,  Professor  171 
Miami  University ,  O.  : — An  ex- 
cellent G^omftlvy  for  the  pupil,  and 
that  is  certainly  what  we  want.  It 
is  written  with  sufficient  fulness 
and   is   suitable   for   the   kind  of 


pupils  we  have,  possessed  of  hu- 
man nature.  It  is  direct  and 
simple  and  easily  intelligible. 

John  R.  French,  Professor  in 
Syracuse  University : — I  am  pleased 
with  it.  It  seems  to  me  admirably 
adapted  to  use  in  the  recitation 
room. 

Daniel  Carhart,  Professor  in  the 
Western  University  of  Pennsyl- 
va7iia  : — The  definitions,  often  ac- 
companied by  illustrations,  are 
made  clear  ;  the  exercises  seem  to 
be  judiciously  chosen,  and  the 
queries  introduced  occasionally 
are  well  selected.  The  chapter 
on  Conic  Sections  will  prove  help- 
ful. The  work  as  a  whole  is  com- 
mendable. 

E.  L.  Caldwell,  Morgan  Park 
Academy,  III.  : — I  find  in  them  the 
best  results  of  modern  research 
combined  with  rigid  exactness  in 
definition  and  demonstration. 

Mary  C.  Noyes,  Lake  Erie  Sem- 
inary, Painesville,  O.  : — It  seems 
to  me  far  better  to  give,  as  Gillet 
does,  some  of  the  propositions  in 
the  form  of  original  work  for  the 
pupil  instead  of  giving  full  demon- 
strations for  all  of  them.  The 
metric  exercises  given  are  calcu- 
lated to  give  pupi.s  clear  ideas  of 
the  subject  and  of  its  applications. 


Mathematics 


Keigwin's  Elements  of  Geometry. 

By  Henry  W.  Keigwin,  Instructor  in  Mathematics,  Norwich  (Ct.)  Free 
Academy,     iv -4-227  pp.     i2mo.     $1.00. 

This  little  book  is  a  class-book,  and  not  a  treatise.  It  cov- 
ers the  ground  required  for  admission  to  college,  and  includes 
in  its  syllabus  the  stock  theorems  of  elementary  geometry.  It 
is,  however,  out  of  the  common  run  of  elementary  geometries 
in  the  following  particulars  : 

1.  The  early  propositions,  and  a  few  difficult  and  funda- 
mental propositions  later,  are  proved  at  length  to  furnish 
models  of  demonstration. 

2.  The  details  of  proof  are  gradually  omitted,  and  a  large 
part  of  the  work  is  developed  from  hints,  diagrams,  etc. 

3.  The  problems  of  construction  are  introduced  early,  and 
generally  where  they  may  soon  be  used  in  related  propositions. 


Oren  Root,  Professor  in  Hamil- 
ton College,  N.  V. : — I  like  the 
book,  especially  in  that  it  gives 
"  inventional  geometry"  while 
giving  the  fundamental  propo- 
sitions. Geometry  is  taught  very 
largely  as  if  each  proposition  were 
an  independent  ultimate  end. 
Pupils  do  not  grasp  the  interlock- 
ing relations  which  run  on  and  on 
and  on  unendingly.  Mr.  Keig- 
win's book,  compelling  pupils  to 
use  what  they  have  learned  of  re- 
lations, must  help  to  prevent  this. 

C.  L.  Gruher,Fo.  Normal Sc/iool, 
Kutztown  : — The  method  of  the 
book  is  an  excellent  one,  since  it 
gradually  leads  the  student  to  de- 
pend in  a  measure  upon  himself 
and  consequently  strengthens  and 
develops  his  reasoning  powers  in 
a  manner  too  often  neglected  by 
teachers  of  the  present  day.  It 
gives  neither  too  little  nor  too 
much. 

W.  A.  Hunt,  High  School,  Den- 
ver, Colo.  : — It  does  not  do  for  the 
pupil  what  he  should  do  for  him- 
self. With  strong  teaching,  the 
book  is  just  what  is  needed  in 
preparatory  schools. 


Miss  Emily  F.  Webster,  State 
Nor77ial  School,  Oshkosh,  Wise: — 
At  the  first  I  looked  upon  the 
book  as  very  small,  but  I  now  con- 
sider it  very  large,  for  it  is  per- 
fectly packed  with  suggestions 
and  queries  which  might  easily 
have  extended  the  book  to  twice 
its  present  size  had  the  author 
seen  fit  to  elaborate,  as  so  many 
authors  do  ;  but  in  not  doing  so 
lies  one  of  the  finest  features  of 
the  book,  as  much  is  thus  left  for 
the  student  to  search  out  for  him- 
self. The  original  exercises  are 
fine  and  in  some  cases  quite  un- 
usual. The  figures  are  clear  and 
the  lettering  is  economical,  some- 
thing which  is  by  no  means  com- 
mon, and  much  valuable  time  is 
wasted  by  repeating  unnecessary 
letters  in  a  demonstration.  Dem- 
onstrations are  made  general, 
which  is  an  advantage,  for  it  is 
often  difficult  to  induce  pupils  to 
do  so  when  the  author  has  failed 
to  set  them  the  example. 

George  Buck,  Dayton  {0,)High 
School . — I  am  highly  pleased  with 
it  and  commend  its  general  plan 
most  heartily. 


Mathematics 


Newcomb*s  School  Algebra. 

By  Simon  Newcomb,  Professor  of  Mathematics  in  the  Johns  Hopkins 
University,  x  4-  294  pp.  i2mo.  95  cents.  (A'<?y,  95  cents.  Answers^  10 
cents.) 

Newcomb's  Algebra  for  Colleges. 

By  Simon  Newcomb,  Professor  in  the  Johns  Hopkins  University.  /Revised. 
xiv  +  546  pp.     i2mo.     $1.30.     (A'^j,  $1.30.     Answers,  10  c&nis,) 

This  book  is  intended  to  cover  the  course  in  algebra  pursued 
by  students  in  our  colleges,  scientific  schools,  and  best  prepara- 
tory schools,  with  such  extensions  as  may  seem  necessary  to 
afford  an  improved  basis  for  more  advanced  studies. 

Newcomb*s  Elements  of  Geometry. 

By  Simon  Newcomb,  Professor  of  Mathematics  in  the  Johns  Hopkins 
University.     Revised,     x  +  399  pp.     i2mo.     $1.20. 

The  first  twenty-four  pages  are  devoted  to  a  drill  in  the  ele- 
ments of  geometric  relation,  with  a  number  of  exercises.  Long 
and  careful  attention  has  led  to  the  adoption  of  definitions 
less  abstract  than  those  usually  found  in  our  elementary  books. 

Newcomb's  Elements  of  Plane  and  Spherical  Trigo- 
nometry.    (With  Five-place  Tables.) 

With  Logarithmic  and  other  Mathematical  Tables  and  Examples  of  their 
Use  and  Hints  on  the  Art  of  Computation.  By  Simon  Newcomb,  Pro- 
fessor of  Mathematics  in  the  Johns  Hopkins  University.  Revised,  vi  + 
168  -f  vi  4-  80  +  104  pp.     8vo.     $i.6o. 

Elements  of  Trigonometry  separate,     vi  +  168  pp.     $1.20. 

Mathematical  Tables,  with  Examples  of  their  Use  and  Hints  on  the  Art 
of  Computation,     vi  -f  80  +  104  PP-     $1.10. 

The  Tables,  which  are  to  five  places  of  decimals,  are  regu- 
larly supplied  to  the  United  States  Military  Academy  and  to 
Princeton   University  and  Yale  University  for  the  entrance 
examinations. 
Newcomb's  Essentials  of  Trigonometry. 

Plane  and  Spherical.  With  Three-  and  Four-place  Logarithmic  and 
Trigonometric  Tables.  By  Simon  Newcomb,  Professor  of  Mathematics 
in  the  Johns  Hopkins  University,     vi  -I-  187  pp.     i2mo.     $1.00. 

This  work  is  much  more  elementary  in  treatment  than  the 
larger,  and  contains  some  practical  applications  of  the  formu- 
las to  surveying.  It  includes  Three-  and  Four-place  Loga- 
rithmic and  Trigonometric  Tables. 


Mathematics 


Newcomb's  Plane  Geometry  and  Trigonometry. 

By  Simon  Newcomb,  Professor  of  Mathematics  in  the  Johns  Hopkins 
University,     viii  4-335  pp.     i2mo.     $1.10. 

This  book  is  made  up  of  the  plane  portion  of  the  "  Ele- 
ments of  Geometry,*'  the  plane  portion  of  the  "  Essentials  of 
Trigonometry,"  and  the  four-place  tables  of  the  lattero 
Newcomb's  Elements  of  Analytic  Geometry. 

By  Simon  Newcomb,  Professor  of  Mathematics  in  the  Johns  Hopkins 
University,     viii  +  357  pp.     i2mo.     $1.20. 

This  work  is  adapted  to  both  those  who  expect  to  apply  the 
subject  to  practical  problems  in  Mechanics  and  Engineering, 
and  also  to  those  who  desire  to  make  a  special  study  of  ad- 
vanced mathematics.  The  opening  chapter  of  Part  I  con- 
tains a  summary  of  the  new  ideas  which  the  student  is  now  to 
associate  with  the  use  of  algebraic  language.  The  rest  of  this 
part  corresponds  closely  to  the  usual  college  course  in  plane 
analytic  geometry,  but  is  so  arranged  that  a  practical  course 
may  be  made  up  by  omitting  certain  sections  and  adding  Part 
II,  which  treats  of  geometry  of  three  dimensions.  The  sec- 
tions omitted  in  the  practical  course,  together  with  Part  III, 
form  an  introduction  to  modern  projective  geometry,  a  course 
of  reading  in  which  is  outlined. 

Newcomb's  Elements  of  the  Differential  and  Integral 

Calculus. 

By  Simon   Newcomb,    Professor  of  Mathematics  in  the  Johns  Hopkins 
University.     xii-|-307pp.     i2mo.     $1.50. 

This  work  presents  a  complete  outline  of  the  first  principles 
of  the  subject  without  going  into  developments  and  applica- 
tions further  than  is  necessary  to  illustrate  the  principles. 
An  attempt  has  been  made  to  lessen  the  logical  difficulties 
which  the  beginner  meets  with  by  presenting  the  elements  of 
the  subject  in  a  rigorous   mathematical  form. 

Phillips  and  Beebe's  Graphic  Algebra.  Or  Geometrical 
Interpretations  of  the  Theory  of  Equations  of  One  Un- 
known Quantity. 

By  A.  W.  Phillips  and  W.  Beebe,  Assistant  Professors  of  Mathematics 
in  Yale  College.     Revised  Edition.     156  pp.     8vo.     $1.60. 


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